Charge continuity equation

This section describes the numerical techniques used for solving the set of differential equations that model the electrodiffusion of ions in solution. The method has historically been called the Poisson-Nernst-Planck (PNP) method because it is based on the coupHng of the Poisson equation with the Nernst-Planck equation. The basic equations used in the PNP method include the Poisson equation (Eq. [18]), the charge continuity equation (Eq. [55]), and the current density of the Nemst-Planck equation (Eq. [54]). 

In this annealing technique, the scaling method was applied for two scales. Scales are defined based on different observable structures at different resolutions of SEM and their statistical information. The annealing stochastic reconstruction and the direct simulation of the charge continuity equation are performed in the following subsections. 

The electric conduction efficiency (%), obtained by direct simulation of the charge continuity equation in 500x (%,5oo) and 5000x (rik,50oo) resolutions, can be generalized by its product in a single micrometric efficiency parameter rimUro- ilk>soo >5,ooo)- In this way, the electric conduction efficiency of the nano-metric structure can be estimated by ... 

A device model to describe two-carrier structures is basically similar to that used for one carrier structures except that continuity equations for both earner types are solved. The additional process that must be considered is charge carrier recombination. The recombination is bimolecular, R=y(np), where the recombination coefficient is given by 43)... 

Essentially what has enabled us to rewrite the matrix element in this form is the fact that the sources of the electromagnetic potential, that is, the currents of the charged particles obey the continuity equation d j x) — 0. 

It is noticed that taking the divergence of this last equation leads to conflict with the continuity equation (eqn. 1.7) for electric charge,... 

Based on the quantum mechanical equation of motion, the continuity equation actually defines the relation between the charge density p(r, t) and the current density j(r,t) at a given time,... 

Like the performance of chemical reactors, in which the transport and reactions of chemical species govern the outcome, the performance of electronic devices is determined by the transport, generation, and recombination of carriers. The main difference is that electronic devices involve charged species and electric fields, which are present only in specialized chemical reactors such as plasma reactors and electrochemical systems. Furthermore, electronic devices involve only two species, electrons and holes, whereas 10-100 species are encountered commonly in chemical reactors. In the same manner that species continuity balances are used to predict the performance of chemical reactors, continuity balances for electrons and holes may be used to simulate electronic devices. The basic continuity equation for electrons has the form... 

Equation (73) is a generalization of the continuity equation for electric charge. Indeed, the left-hand side is the standard Eq. (13), which obtains when... 

The total current (I) flowing through such a cell is composed of displacement (7C) and conduction contributions (/R), i.e., we neglect inductive effects and can concentrate on capacitive and resistive elements. For mechanistic considerations we consider current densities, which in the quasi one-dimensional (laterally homogeneous) case are connected with the current via i = Ha (a = area), while in the general case this connection reads i = dl/da (a area vector, d/da. gradient operator with respect to the a-coordinates).4 The continuity equation of the charge density (P)... 

The effect of uncompensated charge may be discussed by reference to Fig. 8. Consider a charge density <7 at the surface in a very narrow film of thickness S. The potential distribution, as before, is determined by the Poisson equation, the continuity equations, and the charge balance. 

The electric currents and charges are interrelated by the continuity equation... 

The forth equation in (8.62) can be obtained by applying the divergence operator V to both sides of the first equation and allowing for the continuity equation for the extraneous currents and charges. In particularly, the Helmholtz equations (8.31) can be written as follows... 

Current density is a measure of the density of flow of a conserved charge. The equation governing the distribution of potential and current flow in electrolyte can be derived from the continuity equation, charge conservation. The divergence of the current density is equal to the negative rate of change of the charge density [6] ... 

The surface concentration of free charge carriers can be determined from the solution to the continuity equation, eq. 5.124, under the steady-state approximation and with suitable boundary conditions, such as Js= s % (Emeline et al., 2003),... 

It was also assumed that the light irradiance I obeys exponential decay characteristics due to absorption of light by the solid photocatalyst. To describe the concentration of surface charge carriers for the above model, the continuity equation (eq. 5.124) was rewritten under steady-state conditions as expressed by eq. 5.128... 

Application of suitable boundary conditions to the solution of the continuity equation for the charge-carrier concentration s and application of eq. 5.131 yields the very complex expression 5.132 for the quantum yield 4>, which arises more from mathematics than from the physics of the problem (for a derivation and definition of some of the parameters used, see Emeline et al, 2003). 

As with all optoelectronic semiconductor devices, the current-voltage characteristics of an organic solar cell can be modelled using the continuity equations for each charge type. For electrons in the steady state we have... 

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. 

Then what is the source of /iM in electromagnetic theory Are there restrictions on A that should also apply to /iM The answer is yes —it is the restriction of gauge invariance in order to yield a unique representation for the electric and magnetic field variables. Additionally, gauge invariance is the necessary and sufficient condition for the existence of conservation laws in the formalism—in this case the requirement of the conservation of electrical charge [13]. The latter follows from the continuity equation,... 

A solution of Eq. (7) that is compatible with the continuity equation, om/m = 0, which in turn leads to the conservation of charge, is... 

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