10 Dielectric Properties Lecture 9

K Asokan

Learning Outcomes:

From this module students may get to know about the following:

1. Detailed study ferroelectricity and dipole theory of ferroelectricity.
2. The classification of ferroelectric materials in tartrate, dihydrogen phosphates and a5rsenates of alkali metals and oxygen octahedron group.

    9.1 Ferroelectricity:

Below a certain temperature it is found that some materials spontaneously acquire an electric dipole moment. By analogy with the magnetic case these materials are called ferroelectrics. Just as with ferromagnets these crystals exhibit a hysteresis curve P versus E and this can be explained by a domain hypothesis. These domains are quite easy to observe with polarized light in some materials.

The transition to the ferroelectric state is a cooperative phenomenon which is accompanied by specific heat anomaly or by a latent heat and it appears that at the transition temperature the crystal lattice spontaneously distorts to a more complicated structure which possesses a permanent electric dipole moment.

There are three types of crystal structure which exhibit Ferroelectricity: (1) Rochelle salt structure, typified by Rochelle salt, NaK (C4H4O6).4H2O, (2) the perovskite group, consisting mainly of titanates and niobates, of which barium titanate, BaTiO3 has been most extensively studied one, and (3) the dihydrogen phosphates and arsenates, e.g., KH2PO4(‘K D P’).

In ferroelectric materials the electric flux density D is not determined uniquely by the applied field but depends upon the previous history of the material. Just as ferromagnetics have regions with aligned magnetic moments, so also in ferromagnetics there are large regions which are characterized by the alignment of the electric field. For this region the electric field turns the whole of such regions in the direction of the field and overcomes the thermal agitation that tends to scatter the electric dipoles in the different directions. The aligned electric fields of such regions of a ferroelectric combiner with the external electric field increasing the flux density thousands of times. The charge of a condenser is increased the same number of times when a ferroelectric is used instead of air. The lower the temperature and the stronger the electric field the more predominant is the effect of the latter over the random thermal agitation. In sufficiently strong fields, the electric dipoles of all regions of the ferroelectric are practically in the direction of the field. This produces the greatest possible charge density. Condensers made of BaTiO3 and other ferroelectric materials concentrate considerable quantity of electric energy with in a small space as ferrites concentrate magnetic energy. When an electric field is applied to a specimen of a ferroelectric crystal, the polarization first rises rapidly with applied field to a value above which the dependence is linear. Linear extrapolation to zero field gives Ps, the saturation or spontaneous polarization. On subsequently reducing the field to zero, residual polarization Pr remains. The field to reduce the polarization to zero is called coercive field and represented by Ec. The existence of a dielectric hysteresis loop in a dielectric material implies that the substance possesses a spontaneous polarization and the value of Ps (depending upon the shape of hysteresis loop) depends upon a number of factors such as the dimensions of the specimen, the temperature, the texture of the crystal, and the thermal and electrical properties of the crystal.

The hysteresis loop of a ferroelectric material changes its shape as the temperature is increased. The height and width decreases with increase of temperature. At a certain temperature known as ferroelectric Curie temperature, the loop merges in to a straight line and the ferroelectric behavior of the material disappears.

Table 1 Properties of some ferroelectric materials at room temperature

Spontaneous polarisation in a ferroelectric material disappears above the Curie temperature. If the temperature is below the Curie temperature, the polarisation or dielectric constant is not a linear function of the field and εris not a constant and so the equation P=Eε0 (εr- 1) cannot be applied directly. In such cases εrfor a ferroelectric material may be defined for the virgin curve as a differential quantity.

We know that the internal field

Fig 1 (a) Hysteresis loop ferroelectric materials, (b) P-E relation above Curie temperature

If [(1 – Nαβ/ε0 )] = 0, one gets a non-vanishing solution for P and therefore there exists the possibility of spontaneous polarization.

We may conclude now that with [1 – Nαβ/ε0 ] = 1, the dielectric constant will become infinite and the substance will become spontaneously polarised. β = 1/3 for cubic structure.

9.2 Dipole theory of Ferroelectricity:

The interaction between adjacent dipoles in a ferroelectric materials is large and a dipole moment has a tendency to align itself in a direction parallel to that of it’s neighbour. Assuming that the permanent dipoles are responsible for spontaneous polarisation, we can write.

P = P0 =Nα0Ei= Nμm L(a)

For higher temperature,

and is known as ferroelectric curie temperature.

This equation is known as Curie-Weiss law and we have shown with the help of simple theory that a ferroelectric material obeys this equation at sufficiently large temperatures.

The Curie point for Rochelle salt is about 240 C and this substance has a very narrow temperature range of about 400 C within which it is ferroelectric.

We shall now show that spontaneous polarisation is possible in a ferroelectric material below the Curie temperature.

We know that,                                                               P = Ps = NµmL (a)                                                  ……….(6)

For very low temperatures, a is a large (or for high electric fields) and hence L (a) → 1. The polarisation is called the saturation polarisation Ps.

When a is large, L (a) = 1 and Nµm= Ps is known as saturation polarization.

Equation (6) becomes

Spontaneous polarization exists if there is a non-vanishing solution for P in equation (7) with the applied field E = 0.

Hence for spontaneous polarization.

of spontaneous polarization decreases and at very high temperatures , there is no spontaneous polarization, because the straight line does not intersect the L(x) curve. Measured values of are Rochelle salt2.1:BaTIO3 0.044; KH2PO4 0.37.

The spontaneous polarization, which is the most characteristic property of a ferroelectric material, usually vanishes above a certain temperature Tc called the ferroelectric curie temperature. In the ferroelectric region, i.e., below Tc, the dielectric constant is evidently a function of the field strength and is no longer a constant. One can of course define differential relative dielectric constant on the basis of equation (1)

Generally the derivative  is taken for the virgin curve at the origin. The dielectric constant r, as defined by equation (11) may become extremely high as the temperature approaches the ferroelectric curie temperature, as evident from figure. showing the variation of dielectric constant r as a function of temperature in BaTIO3 ceramics. The chain dotted curve pertains to a field of 110000 volt/m while the solid curve refers to a field of 5600 volt/m. It may be seen that in the ferroelectric region, i.e., in the temperature range below Tc the dielectric constant r is a function of field,r being higher for higher values of field. However, for temperature exceeding the ferroelectric curie temperature Tc, r does not vary with field.

For temperature exceeding the ferroelectric Curie temperature, the variation of absolute dielectric constant with temperature given by the Curie-Weiss law.

    9.3 Classification of Ferroelectric Materials

Ferroelectric materials may be classified into the following three groups depending on their chemical composition and structure;

a)      Tartrate group

b)      Dihydrogen phosphates and arsenates alkali metals

c)      Oxygen octahedron group.

Table (2) indicates some of the materials which possess ferroelectric properties.

9.3.1Tartrate Group

A typical example of ferroelectric material of this group is Rochelle salt, which is the sodium-potassium salt of tartaric acid NaK (C4H4O6). 4H2O. This material was probably the first solid known to exhibit ferroelectric properties. This material has the unique property that it is ferroelectric only in the temperature range extending from -18 °C to 23 °C. Thus the material has two transition regions instead of one.

Figure….  shows  the  variation  of  spontaneous  polarisation  Ps  of  Rochelle  salt  with temperature. Other materials belongings to this group of ferroelectric materials are those in which a part of Na in the Rochelle salt has been replaced by NH4, Rb or Ti.

9.3.2 Dihydrogen Phosphates and Arsenates of Alkali Metals

A  typical  example  of  this  category is  KH2PO4.  Figure  ….shows  the  spontaneous  polarization  vs temperature curve of this material. In this case, there is only one Curie temperature namely 123 K.

    9.3.3 Oxygen Octahedron Group

Probably the best known ferroelectric material is BaTiO3; it is a representative of the so called oxygen octahedron group of ferroelectric materials. Above the Curie temperature (120 °C), BaTiO3 corresponds to the cubic structure presented in figure. In this structure, Ba2+ ions occupy the corners of a cube; the centres of cube faces are occupied by O2- ions. The oxygen ions form an octahedron, at the centre of which Ti4+ ion is located. The Ti4+is considerably smaller than the space which is available inside the oxygen octahedron.

It thus brings with a high ionic polarizability for two reasons: (i) It has a charge of 4e and, (b) It can be displaced over a relatively large distance. We shall that this may be explanation for the occurrence of spontaneous polarization in BaTiO3.

There is a intimate relationship between the ferroelectric properties and the atomic arrangement in ferroelectric materials. Above 120 °C, BaTiO3 has the cubic structure as in figure. When the temperature is lowered through the critical temperature of 120 °C, the material becomes spontaneously polarized and at the same time the structure changes. The direction of spontaneous polarization may lie along any of the cube edges, giving a total 6 possible directions for spontaneous polarization. Along the direction of spontaneous polarization of a given domain, the material expands, whereas perpendicular to the polarization direction it contracts. Thus, the material is no longer cubic, but corresponds to a so called tetragonal structure. BaTiO3 has two more transition temperatures; One at 5 °C, where the spontaneous polarization changes its direction from one of the cube edges to a direction corresponding to a face diagonal in figure. and one at -80 °C where the spontaneous polarization changes from a direction corresponding to a face diagonal to one along a body diagonal. Associated with each of these ferroelectric transitions is a change in the crystal structure of the material. These three transition temperatures are reflected in the dielectric constant and in the spontaneous polarization of the material, as they may be seen in figure.

The spontaneous polarization represented in figure was measured along a cube edge over the whole temperature range. Thus the magnitude in the range between 193 K and 278 K is obtained by multiplying the value given in figure by √2 (Ps in that region is directed along a face diagonal). Similarly, to obtain the magnitude of Ps in the region below 193 K, one should multiply the value in figure by √3 (Ps directed along body diagonal in this case).

Summary:

In this chapter we had a detailed description of ferroelectricity and dipole theory of ferroelectricity. The different classification of ferroelectric materials in tartrate, dihydrogen phosphates and a5rsenates of alkali metals and oxygen octahedron group.

    References:

1. Werner Känzig (1957). “Ferroelectrics and Antiferroelectrics”. In Frederick Seitz, T. P. Das, David Turnbull, E. L. Hahn. Solid State Physics 4. Academic Press. p. 5. ISBN 0-12-607704-5.
2. M. Lines & A. Glass (1979). Principles and applications of ferroelectrics and related materials. Clarendon Press, Oxford.ISBN 0-19-851286-4.
3. See J.  Valasek  (1920).  “Piezoelectric  and  allied  phenomena  in  Rochelle  salt”. Physical Review 15:537.Bibcode:1920PhRv…15..505.. doi:10.1103/PhysRev.15.505.and J. Valasek (1921).  “Piezo-Electric  and  Allied  Phenomena in Rochelle Salt”. Physical Review 17 (4): 475.Bibcode:1921PhRv…17..475V. doi:10.1103/PhysRev.17.475.
4. Chiang, Y. et al.: Physical Ceramics, John Wiley & Sons 1997, New York
5. Safari, Ahmad (2008). Piezoelectric and acoustic materials for transducer applications. Springer Science & Business Media. p. 21.ISBN 0387765409.

    References and Suggestive Readings

1.  A. S. Sidorkin (2006). Domain Structure in Ferroelectrics and Related Materials. Cambridge University Press. ISBN 1-904602-14-2.
2. Karin M Rabe, Jean-Marc Triscone, Charles H Ahn (2007). Physics of Ferroelectrics: A modern perspective. Springer. ISBN 3-540-34591-4.
3. Julio A. Gonzalo (2006). Effective Field Approach to Phase Transitions and Some Applications to Ferroelectrics. World Scientific.ISBN 981-256-875-1

    Web Links

1.  A useful starter on ferroelectrics

2. A ferroelectrics research group at Stony Brook University

3. http://www.britannica.com/science/ferroelectricity

4. http://www.slideshare.net/researcher1234/ferroelectric-and-piezoelectric-materials

Additional Topics to be studied

1) History of Ferroelectricity

History of Ferroelectricity

It all began with a salt in La Rochelle, a small but important city at the south-west coast of France. Jehan Seignette, born in 1592, a militant protestant, succeded to run a pharmacy in spite of serious obstacles opposed to him by the clerics. One of his sons, Pierre, born in 1623, became a medical doctor at the University of Montpellier and his younger brother Elie, born in 1632, took over his father’s business.  In  these  years  pharmacy  consisted  mainly  in  extracting  plants  and  distilling  essences. Apparently purgatives such as “folia sennae” introduced to Europe by Arab medical men in the early middle age, played an important role. Because of unpleasent side effects patients were very reluctant to take them. This was the reason Dr. Seignette suggested to his brother to look for some mineral drugs or to make some. Although “mineralia” were in use for curing various diseases in eastern countries already 2000 years B.C., Pierres idea was decisive. As the result of hard work Elie came out with a salt in approximately 1665, which he called “sel polychreste” derived from the greek πολυχρηστοσ, which means a salt of various utilities. It was a real creation and the way he produced the salt was kept secret for ever it seems. Only 65 years later the French pharmacist and chemist Simon Boulduc in Paris found out by analysis that sel polychreste must be “some soda”. It is likely that Elie Seignette started from crème of tartrate (potassium hydrogen tartrate) he obtained from wines – so famous and abound in the Bordeaux region – and soda which makes the tartrate soluble in water. The “sel polychreste” – or Rochelle salt as we know it today – conquered the market in France, especially in Paris and it was in widespread use for more than two centuries as a mild drug.

In the nineteenth century – nearly 200 years after its discovery – the physical properties of Rochelle salt began to excite interest. In 1824 David Brewster had observed the phenomenon of pyroelectricity in various crystals, among which was Rochelle Salt but perhaps the first systematic studies were those of the brothers Pierre and Paul-Jacques Curie in 1880. This classic work established unequivocally the existence of the piezoelectric effect and correctly identified Rochelle Salt and a number of other crystals as being piezoelectric. They also noticed that Rochelle salt was by far more active than quartz for instance and all the rest of the crystals they investigated. But the fascinating dielectric features of Rochelle salt escaped them. Thomas Alva Edison was maybe the first who used its piezoelectrical effect in a commercial application in 1899 – the phonograph. However, his invention was just a curiosity and far too expensive. At this time Rochelle salt was of pure academic significance. During World War I, however, physicists and electrical engineers showed an increasing interest in its physical properties mainlybecause of its unusually high piezoelectric moduli. At the beginning of the war 1914-18 A.M. Nicholson in the USA and Paul Langevin in France began to perfect independently an ultrasonic submarine detector. Their transducers were very similar: a mosaic of thin quartz crystals glued between two steel plates (the composite having a resonant frequency of about 50 KHz), mounted in a housing suitable for submersion. Working on past the end of the war, they did achieve their goal of emitting a high frequency “chirp” underwater and measuring depth by timing the return echo. The strategic importance of their achievement was not overlooked by any industrial nation, however, and since that time the development of sonar transducers, circuits, systems, and materials has never ceased.

Petrus Josephus Wilhelmus Debye, or Peter Debye as we know him today, professor of theoretical Physics at the University of Zürich had carefully observed the work on piezoelectricity and in 1912 he came up with an idea. To explain the results he knew he brought forth the hypothesis that a certain class of molecules carry a permanent electric dipole moment in analogy to the magnetic moment of the atoms of paramagnetic substances. Following Langevin’s theory of paramagnetism Debye gave the equation (ε-1)/(ε+2)=a+b/T, where a is proportional to the density of the substance and b to the square of the electric dipole moment. This relation was perfectly confirmed later in many cases.

Debye came to a further conclusion. According to his relation for a critical temperature TK=b/(1-a) the dielectric constant reaches infinity. Therefore, he proposed TK to be the analogue to the Curie temperature of a ferromagnet. For temperatures lower than TK a permanent dielectric polarisation ought to be expected even in the absence of an electric field. To his knowledge, he said, no such a phenomenon had been observed so far. The basic feature of ferroelectricity was anticipated, however! Erwin Schrödinger in his “Habilitations-Schrift” submitted at the University of Vienna late in 1912, the year Debye published his „Vorläufige Mitteilung“, went a step further. He elaborated on Debye’s simple model and tried to extend it to solids. If this could be done successfully, Schrödinger speculated, then all solids should become “ferroelektrisch” at a sufficiently low temperature. So, in fact, the term ferroelectric or ferroelectricity was coined by Schrödinger as early as 1912!