Space-charge density

The average space charge density (per unit volume) becomes... 

Often, the concept of (two-dimensional) surface or (three-dimensional) space charge is employed. Here it is assumed that the charge is distributed in a continuous fashion (smeared out) over the surface S or volume V. Surface and space charge can be described in terms of surface-charge density = dQldS or space-charge density Qy = dQldV, which may either be constant or vary between points. 

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. 

Figure 9 shows the electrical potential and the space charge density profiles for concentrations c" = 1000 mM and c = 20 mM and Galvani potential difference f =... 

FIG. 9 Simulated electrical potential and space charge density profiles at the water-1,2-DCE interface polarized at/= 5 in the absence (a) and in the presence (b) of zwitterionic phospholipids. The supporting electrolyte concentrations are c° = 20 mM and c = 1000 mM. The molecular area of the phospholipids is 150 A, and the corresponding surface charge density is a = 10.7 xC/cm. The distance between the planes of charge associated with the headgroups is d = 3 A. 

At the final concentration c, the potential j)k at distance r is given not only by the potential of the ion, )°ky but also by the potentials of the surrounding ions. Debye and Hiickel assumed that a spherical ionic atmosphere of statistically prevailing ions with opposite charge forms around each ion, giving rise to the potential ipa. Thus xpk = ipk + The potential of the space charge density p is given by the Poisson equation... 

The space charge density is given by the sum of the charges of all the ions present ... 

Consider a plane metal electrode situated at z = 0, with the metal occupying the half-space z < 0, the solution the region z > 0. In a simple model the excess surface charge density a in the metal is balanced by a space charge density p(z) in the solution, which takes the form p(z) = Aexp(—kz), where k depends on the properties of the solution. Determine the constant A from the charge balance condition. Calculate the interfacial capacity assuming that k is independent of a. 

FIGURE 5.4 (See color insert following page 302.) Coordinate and momentum space charge densities of H2 molecule illustrating the hond directionality principle. Isosurfaces from 0.04 to 0.01 a.u. are plotted for the coordinate space charge density (a). Isosurfaces from 1.0 to 0.01 a.u. are plotted for the momentum space charge density (h). 

FIGURE 5.6 (See color insert following page 302.) Momentum and coordinate space charge density profiles for the reaction path from HNC to HCN. 

Yet another indirect connection between momentum and coordinate space charge densities is derived via a quantity called the Shannon information entropy... 

It should be observed that Eq. (7) is of a Proca type, here being due to generation of a space-charge density p in vacuo (free space). Such an equation can describe a particle with the spin value unity [31]. 

Here, p is the pressure and p is the space charge density, which for the equilibrium ionic fluid under consideration assumes the form... 

Some typical stationary voltage-current VC curves along with the ionic concentration, space charge density, and the electric field intensity profiles for an intermediate voltage range are presented in Fig. 5.3.1. The appropriate profiles are constructed using a numerical solution of the system (5.3.1), (5.3.5). The essence of the numerical procedure employed for this and similar problems discussed in due course is as follows. 

It is observed from Fig. 5.3.2c that in the modified TMS model (5.3.61) (5.3.63) permselectivity (t ) remains essentially constant in a wide range of voltages. The corresponding VC curves (Fig. 5.3.2a,b) are similar in shape to those for an ideal permselectivity membrane. This is also true regarding the space charge density and profiles of the other fields in the depletion layer 0 < x < 1. 

The space charge density is g = (F/Vm)-(Nd+ -Nd-), and their characteristic width (Le., the Debye-HGckel length, which is an equilibrium property and independent of the d-mobilities) is obtained as... 

The creation of a large space-charge density in the semiconductor (and a corresponding increase in capacitance) can cause the bands to become unpinned since changes in applied potential can now occur across the Helmholtz layer. 

In continuum notation, this relation would constitute one form of Poisson s equation of electrostatics. The continuum forms of E(x) and V (x) are valid if the charge density planes are so close together that over small regions of space the charge density can be viewed as a continuous function p(x) of position x. [The local space charge density p(x) has units of Coulombs m-3]. In such cases, the sums in eqns. (37) and (40) for E (x) and V (x) can be approximated by integrals to give... 

Note that the surface-charge field E(0) has the same sign as the surface-charge density r0. The field E(x) at position x in the oxide for the present case of a uniform space charge density follows readily from eqn. (41), viz. 

The present qualitative deductions can be supplemented by a more quantitative treatment which has been presented in the literature [ 21 ]. One can refer to that development for the predicted functional dependences of the current on space-charge density. 

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