Nonlinear dielectric

Polarization which can be induced in nonconducting materials by means of an externally appHed electric field is one of the most important parameters in the theory of insulators, which are called dielectrics when their polarizabiUty is under consideration (1). Experimental investigations have shown that these materials can be divided into linear and nonlinear dielectrics in accordance with their behavior in a realizable range of the electric field. The electric polarization PI of linear dielectrics depends linearly on the electric field E, whereas that of nonlinear dielectrics is a nonlinear function of the electric field (2). The polarization values which can be measured in linear (normal) dielectrics upon appHcation of experimentally attainable electric fields are usually small. However, a certain group of nonlinear dielectrics exhibit polarization values which are several orders of magnitude larger than those observed in normal dielectrics (3). Consequentiy, a number of useful physical properties related to the polarization of the materials, such as elastic, thermal, optical, electromechanical, etc, are observed in these groups of nonlinear dielectrics (4). 

The most important materials among nonlinear dielectrics are ferroelectrics which can exhibit a spontaneous polarization PI in the absence of an external electric field and which can spHt into spontaneously polarized regions known as domains (5). It is evident that in the ferroelectric the domain states differ in orientation of spontaneous electric polarization, which are in equiUbrium thermodynamically, and that the ferroelectric character is estabUshed when one domain state can be transformed to another by a suitably directed external electric field (6). It is the reorientabiUty of the domain state polarizations that distinguishes ferroelectrics as a subgroup of materials from the 10-polar-point symmetry group of pyroelectric crystals (7—9). 

When a strong static electric field is applied across a medium, its dielectric and optical properties become anisotropic. When a low frequency analyzing electric field is used to probe the anisotropy, it is called the nonlinear dielectric effect (NLDE) or dielectric saturation (17). It is the low frequency analogue of the Kerr effect. The interactions which cause the NLDE are similar to those of EFLS. For a single flexible polar molecule, the external field will influence the molecule in two ways firstly, it will interact with the total dipole moment and orient it, secondly, it will perturb the equilibrium conformation of the molecule to favor the conformations with the larger dipole moment. Thus, the orientation by the field will cause a decrease while the polarization of the molecule will cause an... 

Figure 2. Nonlinear dielectric effect constant ( 10 scm7SC 2moF ) of Br(CH2)n, Br at 25 °C. Points represent the experimental results of Chelkowski (19, 20). The solid and dashed lines are calculated with 4>s, = 120 ° and +80 °, respectively, for thefirst and last C-C bonds. (Reproduced with permission from Ref 6. Copyright 1981, American...

We consider a nonlinear dielectric medium in one dimension. The relation between the polarization P and the electric field E is given by ... 

Nonlinear chemometrics methods, 6 53-54 Nonlinear dielectrics, 11 91-92 Nonlinear fracture mechanics, 20 350 Nonlinear interaction, 14 680... 

Directly following the development of the optical laser, in 1961 Frankel et al. [10] reported the first observation of optical harmonics. In these experiments, the output from a pulsed ruby laser at 6943 A was passed through crystalline quartz and the second harmonic light at 3472 A was recorded on a spectrographic plate. Interest in surface SHG arose largely from the publication of Bloembergen and Pershan [11] which laid the theoretical foundation for this field. In this publication, Maxwell s equations for a nonlinear dielectric were solved given the boundary conditions of a plane interface between a linear and nonlinear medium. Implications of the nonlinear boundary theory for experimental systems and devices was noted. Ex-... 

The isotropic coefficient and the anisotropic coefficients b(m> and c(m) can have both bulk and surface contributions and depend on crystal symmetry. The linear and nonlinear dielectric constants of the material, as well as the appropriate Fresnel factors at co and 2co, are incorporated into the constants a, b m) and c(m). Table 3.1 shows the susceptibilities contained in each of these constants. The models of Tom... 

A linear dielectric medium is characterized by a linear relation between the polarization density and the electric field, P = e0xE, where eo is the permittivity of free space and x is the electric susceptibility of the medium. A nonlinear dielectric medium, on the other hand, is characterized by a nonlinear relation between P and E, as illustrated in Figure 4.19. 

Figure 5.7 (e) shows a so-called split post resonator, which allows for a highly precise determination permittivity and loss tangent of dielectric sheet materials [5], In addition, it is also considered to be useful for measurements of nonlinear dielectric films. 

As already discussed above, nonlinear dielectric films can be characterized by planar structures. Resonant planar structures allow - to a certain extend - the determination of the loss tangent. Figure 5.9 shows a resonator assembly which has been used for loss tangent measurements on SrTiOa thin film varactors [16]. 

Figure 5.9 Planar nonlinear dielectric based thin film varactor structure (top) and planar microstrip resonator with capacitive gap. The microwave dielectric properties of the varactor structure are measured in a flipchip configuration of both substrates with the narrow (few microns) capacitive gap of the varactor structure being placed in the center of the large (500 microns) capacitive gap of the resonator structure (from [16]).

The microwave properties of oxide based dielectric bulk material, thin film nonlinear dielectric materials and oxide high temperature superconducting materials were reviewed in this article. In addition, the most important microwave measurement techniques have been discussed. Important future directions of related material research aiming towards further integration both on chip and subsystem level, increase of performance and cost reduction are ... 

In order to compare calculated and experimentally observed phase portraits it is necessary to know very exactly all the coefficients of the describing nonlinear differential Equation 14.3. Therefore, different methods of determination of the nonlinear coefficient in the Duffing equation have been compared. In the paraelectric phase the value of the nonlinear dielectric coefficient B is determined by measuring the shift of the resonance frequency in dependence on the amplitude of the excitation ( [1], [5]). In the ferroelectric phase three different methods are used in order to determine B. Firstly, the coefficient B is calculated in the framework of the Landau theory from the coefficient of the high temperature phase (e.g. [4]). This means B = const, and B has the same values above and below the phase transition. Secondly, the shift of the resonance frequency of the resonator in the ferroelectric phase as a function of the driving field is used in order to determine the coefficient B. The amplitude of the exciting field is smaller than the coercive field and does not produce polarization reversal during the measurements of the shift of the resonance frequency. In the third method the coefficient B was determined by the values of the spontaneous polarization... 

With this background, we have proposed and developed a new purely electrical method for imaging the state of the polarizations in ferroelectric and piezoelectric material and their crystal anisotropy. It involves the measurement of point-to-point variations of the nonlinear dielectric constant of a specimen and is termed scanning nonlinear dielectric microscopy (sndm) [1-7]. This is the first successful purely electrical method for observing the ferroelectric polarization distribution without the influence of the screening effect from free charges. To date, the resolution of this microscope has been improved down to the subnanometer order. 

Here we describe the theory for detecting polarization and the technique for nonlinear dielectric response and report the results of the imaging of the ferroelectric domains in single crystals and thin films using sndm. Especially in a measurement of pzt thin film, it was confirmed that the resolution was sub-nanometer order. We also describe the theoretical res-... 

Nonlinear dielectric imaging with sub- nanometer resolution... 

Figure 16.1 shows the system setup of the sndm using the lc lumped constant resonator probe [4], In the figure, Cs(t) denotes the capacitance of the specimen under the center conductor (the tip) of the probe. Cs (t) is a function of time because of the nonlinear dielectric... 

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