Distribution of space charge

The relation between the spatial potential distribntion and the spatial distribution of space-charge density can be stated, generally, in terms of Poisson s dilferential equation. 

For frequencies between f, and f, the distribution of space charge and potential within the discharge is governed by the manner in which the applied field affects the motions of ions and electrons. Because of their low mobility the positive ions are... 

A similar expression is to be expected for the distribution of electrons in exhaustion boundary layers. The evaluation of these equations is not easy. Therefore, it is desirable to simplify our assumptions concerning the distribution of the electrons and holes in the boundary layer. This simplification is illustrated by Fig. 3. With a suitable choice of the thickness I of the boundary layer, the simplification will satisfactorily approximate the real case. The distribution of space charge in an inundation boundary layer can be similarly calculated, and will be shown below in parentheses. 

Fig. 7.9. PEA output trace showing the distribution of space charge in a sample of PVC film (420 pm thick) without any applied voltage. Reproduced from Maeno and Fukunaga (1996) with permission of the Institute of Electrical and Electronics Engineers, Inc.

Suppose that a semiconductor of thickness L is contacted with an electrode that, hy virtue of a low-energy barrier at the interface, is able to supply an unlimited number of one type of carrier. The current is then limited by its own space charge which, in die extreme case, reduces the electric field at the injecting contact to zero. This is realized when the number of carriers per unit area inside the sample approaches the capacitor charge, i.e. sso/e. It is this number of carriers dial can be transported per transit time ttr=d/fji. Hence, the maximum current is iscL = s,E,QfiF ld. A more rigorous treatment has to take into account the non-uni-form distribution of space charge and, concomitantly, electric field [34]. Starting with Poisson s equation and the continuity equation. 

Distribution of space charges (electrodynamics, condensed matter)... 

FIGURE 5.11. Diagrams of the distribution of space charge and of the director (broken lines) with a change in the polarity of the field (a) at frequencies below the critical frequency and (b) at frequencies above the critical frequency. The directions of fiow are indicated by arrows. 

The expression for V(r), given as Eq. (3.1), follows from the definition of electrical potential, which will be reviewed here. Any distribution of electrical charge creates a potential V(r) in the surrounding space. For an assembly of point charges ). located at positions r, this electrical potential is simply a sum of Coulombic potentials, as given in Eq. (3.2). 

The electrons and nuclei of an atom or molecule (or other system) are charged particles, and therefore create an electrical potential in the surrounding space. Because we normally treat the electrons as a continuous but static distribution of negative charge, with the nuclei forming a rigid framework, the resulting electrostatic potential V(r) is, by Coulombs law ... 

Support for the applicability of this model to an explanation of the Meyer-Neldel rule comes from measurements of space-charge limited currents in anthracene where a correlation (see Fig. 20) has been found between the total density of traps H and the distribution parameter Tc (Owen et al, 1974). It has been shown that this effect is not fortuitous as suggested by some workers... 

Miyaki and Fujimoto and co-workers [16,17] have obtained an even finer distribution of fixed charge groups by casting films from multicomponent block copolymers such as poly(isoprene- >-styrene- >-butadiene- >-(4-vinyl benzyl)dime-thylamine- Msoprene). These films show a very regular domain structure with a 200-500 A spacing. After casting the polymer film, the (4-vinyl benzyl) dimethy-lamine blocks were quatemarized with methyl iodide vapor, and the styrene blocks were sulfonated with chlorosulfuric acid. 

Fig. 5.47 Distribution of net charge in the water molecule (electrostatic charge, calculated with AMI - Chapter 6). Negative to positive red to blue R O Y G B). (a) Slice through the plane of the molecule the contour lines show the decrease in net negative charge, (b) Charge in space this corresponds essentially to the lone pairs, (c) Charge mapped on the van der Waals surface...

Added to these fields, of course, would be any externally applied field, E0, due to charge distributions other than the planes of space charge presently considered. Using the convention of labeling the total electric field existing in the region between charge planes — 1 and as Ejt then we can write... 

Molecules with equal numbers of protons and electrons are not charged. But the distribution of electrons in three dimensions need not be uniform in the molecule with respect to the distribution of the nuclei. In this case, the distribution of negative charge in a molecule may be different from the distribution of positive charge, and the molecule has one or more bond dipole moments, the magnitude of which depends on these distributions. The more uneven the distributions of charge in space, the more polar the molecule is.3... 

Experimental determination of charge density relies mostly on X-ray diffraction although other techniques have been applied in some instances. X-ray diffraction arises from scattering by electrons and therefore carries information on the distribution of electronic charge in real space [10]. The intensity of a Bragg reflection, 7(h), at a given temperature, is proportional to the square of its structure factor,... 

---