Local space charge neutrality

As one example of a coupled-currents theory, the growth of very thick oxide films under local space-charge neutrality conditions will be considered in detail in the following section. 

If local space-charge neutrality can be assumed, which has been justified by extensive numerical calculations [46] for the present case, then we can write... 

Equation (180) for local space-charge neutrality requires that the oppositely charged mobile defect species concentrations have a very special relationship. Solving for C2 gives... 

Since local space-charge neutrality does not hold at the oxide interfaces, the above expression for the current is restricted to the interior zone [28] where local space charge neutrality has been found [46] to be a good approximation. This is illustrated for the case of cation vacancy (or anion interstitial) and electron-hole diffusion by Fig. 17. Thus, the domain of validity is not 0 but instead is 5 < [Pg.75]

To evaluate C](5 ) and C1(L ), we introduce what has been designated [28, 29] the interfacial zone equilibrium approximation the concentration profiles of all charged defect species within the two interfacial regions (0interior zone (5 [Pg.75]

These values for Cs(8 ) and C (L ) are likewise appropriate at the respective edges of zone 2, since defect concentrations are continuous functions of position within the oxide layer. The subscript s can have the value 1 or 2 in the above relations, corresponding respectively to the ionic or electronic species concentrations. Substituting eqn. (196) for Cs(8 ) into the local space-charge neutrality relation [eqn. (180)] gives... 

Next we substitute eqn. (197) into the local space-charge neutrality relation [eqn. (180)] to obtain... 

The next level is that of one-dimensional electro-diffusion with local electro-neutrality in the absence of an electric current. This is the realm of nonlinear diffusion to be treated in Chapter 3. A still higher level of the same hierarchy is formed by the nonlinear effects of stationary electric current, passing in one-dimensional electro-diflFusion systems with local electro-neutrality. A few typical phenomena of this type will be studied in Chapter 4. The treatment of Chapter 4 will lay the foundation for the discussion of the effects of nonequilibrium space charge characteristic of the fourth level to be treated in Chapter 5. 

In 4.4 the theory of 4.2 will be applied to study electro-diffusion of ions through a unipolar ion-exchange membrane, separating two electrolyte solutions. This will include the classical treatment of concentration polarization in a solution layer adjacent to an ion-exchange membrane under an electric current. The validity limits of this theory, set by the violations of local electro-neutrality and caused by the development of a macroscopic nonequilibrium space charge, will be indicated. (The effects of the nonequilibrium space charge are to be discussed at some length in Chapter 5.)... 

A few remarks are due about this feature. The nonuniformity above is a formal expression of breakdown of the local electro-neutrality assumption in concentration polarization, described in the previous chapter. Essentially, this reflects the failure of a description based upon assuming the split of the physical region into a locally electro-neutral domain and an equilibrium double layer where all of the space charge is concentrated. The source of this failure, reflected in the nonuniformity of the corresponding matched asymptotic expansions, is that the local Debye length at the interface tends to infinity as the voltage increases. In parallel a whole new type of phenomena arises, which is not reflected in the simplistic picture above. The... 

Anomalous rectification [3]. Our aim in this section is to show that under certain conditions development of a nonequilibrium space charge may yield, besides the punch through, some additional effects, unpredictable by the locally electro-neutral formulations. We shall exemplify this by considering two parallel formulations—the full space charge one and its locally electro-neutral counterpart. It will be observed that inclusion of the space charge into consideration enables us to account for the anomalous rectification effect that could not be predicted by the locally electro-neutral treatment. Physical motivation for this study is as follows. 

Unambiguous steady-state measurements are carried out only in the trap-free space-charge-limited-current (SCLC) regime (23), when the average transit time (f j of any excess injected carrier across the sample bulk is shorter than the time required for the bulk to locally neutralize the carrier (22). The transit time ( r) has been defined in equation 1. The time to neutralize any excess injected carrier is the bulk dielectric relaxation time, Tr ( r = in which p is the bulk resistivity and e is the bulk dielectric... 

The space charge density p(z) neutralizing the surface charge is related to the local electrolyte concentrations. For a symmetrical indifferent electrolyte p(z) can be written as ... 

Additional sources for potential barriers in ionic systems can be driven by intrinsic ionic processes. As first described by Frenkel, the formation of a net surface charge and a compensating space charge layer relates to the energy differences required to bring various ionic species to a surface [13]. Indeed, while ionic soUds are macro-scopicaUy charge-neutral, local variations in both structure and chemistry lead to internal electrostatic potentials and electric fields. Space charge layers are formed... 

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