Space-charge region

For most electrochemical systems, the separation of charge associated with interfacial regions can be treated simply as contributing to rate constants associated with 

Remember 12.2 Expressions for transport cf electrons and holes in a semiconductor can be treated in terms of gradients of the corresponding electrochemical potential. 

The term (N — No) includes the charge associated with partially ionized mid-bandgap acceptors (which may be a function of applied potential) as well as the completely ionized dopant species (which may have an arbitrary distribution, but is usually assumed to be independent of operating conditions). 

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The primary photochemical act, subsequent to near-uv light (wavelengths <400 nm) absorption by Ti02 particles, is generation of electron—hole pairs where the separation (eq. 3) into conduction band electrons (e g ) and valence band holes (/lyB ) faciUtated by the electric field gradient in the space charge region. Chemically, the hole associated with valence band levels is constrained at... 

The layer is so thick that the space-charge regions at the two interfaces are unimportant and the oxide can be regarded as electrically neutral. 

The photoinduced microwave conductivity signal, on the other hand, can be described by the following integral over the excess minority carriers, to be taken over both the diffusion and the space charge region ... 

An interesting special application has been proposed by Schlichthorl and Peter.31,41 It aims at deconvolution of electrochemical impedance data to separate space charge and surface capacitance contributions. The method relies on detection of the conductivity change in the semiconductor associated with the depletion of majority carriers in the space charge region via potential-modulated microwave reflectivity measurements. The electrode samples were n-Si(lll) in contact with fluoride solution. 

Semiconductor-electrolyte interface, photo generation and loss mechanism, 458 Semiconductor-oxide junctions, 472 Semiconductor-solution interface, and the space charge region, 484 Sensitivity, of electrodes, under photo irradiation, 491 Silicon, n-type... 

According to the model proposed by Verwey and Niessen (1939), an electric double layer is formed at an ITIES, which consists of two ionic space charge regions. As a whole the electric double layer is electrically neutral, so for the excess charge density in the part of the double layer in the aqueous phase, and for the part in the organic phase,... 

Assuming that the two space-charge regions are separated by a layer of solvent molecules (inner layer or mixed solvent layer), the Galvani potential difference can be expressed as the sum of three contributions ... 

FIGURE 32.3 Modified Verwey-Niessen model of an ITIES with an inner layer (shaded area) separating two space-charge regions. 

The lattice gas has been used as a model for a variety of physical and chemical systems. Its application to simple mixtures is routinely treated in textbooks on statistical mechanics, so it is natural to use it as a starting point for the modeling of liquid-liquid interfaces. In the simplest case the system contains two kinds of solvent particles that occupy positions on a lattice, and with an appropriate choice of the interaction parameters it separates into two phases. This simple version is mainly of didactical value [1], since molecular dynamics allows the study of much more realistic models of the interface between two pure liquids [2,3]. However, even with the fastest computers available today, molecular dynamics is limited to comparatively small ensembles, too small to contain more than a few ions, so that the space-charge regions cannot be included. In contrast, Monte Carlo simulations for the lattice gas can be performed with 10 to 10 particles, so that modeling of the space charge poses no problem. In addition, analytical methods such as the quasichemical approximation allow the treatment of infinite ensembles. 

As an example Fig. 6 shows the distribution of the ions for a potential difference of A(j) = 0(00) — 0(—00) = kT/cq between the two bulk phases. In these calculations the dielectric constant was taken as e = 80 for both phases, and the bulk concentrations of all ions were assumed to be equal. This simplifies the calculations, and the Debye length Lj), which is the same for both solutions, can be used to scale the v axis. The most important feature of these distributions is the overlap of the space-charge regions at the interface, which is clearly visible in the figure. 

Unfortunately, at the present time the experimental results for ion-transfer reactions are contradictory, so that it is not possible to verify the predictions of this model. Also, this model is only valid if the rate is determined by the ion-transfer step, and not by transport, and if the concentration of the supporting electrolyte is sufficiently low so that the extension of the space-charge regions is less than the width X of the region where the two solvents mix. These conditions are not always fulfilled in experiments. 

In the lattice-gas model, as treated in Section IV.D above, ion transfer is viewed as an activated process. In an alternative view it is considered as a transport governed by the Nernst-Planck or the Langevin equation. These two models are not necessarily contra-dictive for high ionic concentrations the space-charge regions and the interface have similar widths, and then the barrier for ion transfer may vanish. So the activated mechanism may operate at low and the transport mechanism at high ionic concentrations. 

Samec et al. [15] used the AC polarographic method to study the potential dependence of the differential capacity of the ideally polarized water-nitrobenzene interface at various concentrations of the aqueous (LiCl) and the organic solvent (tetrabutylammonium tetra-phenylborate) electrolytes. The capacity showed a single minimum at an interfacial potential difference, which is close to that for the electrocapillary maximum. The experimental capacity was found to agree well with the capacity calculated from Eq. (28) for 1 /C,- = 0 and for the capacities of the space charge regions calculated using the GC theory,... 

To evaluate the contribution of the SHG active oriented cation complexes to the ISE potential, the SHG responses were analyzed on the basis of a space-charge model [30,31]. This model, which was proposed to explain the permselectivity behavior of electrically neutral ionophore-based liquid membranes, assumes that a space charge region exists at the membrane boundary the primary function of lipophilic ionophores is to solubilize cations in the boundary region of the membrane, whereas hydrophilic counteranions are excluded from the membrane phase. Theoretical treatments of this model reported so far were essentially based on the assumption of a double-diffuse layer at the organic-aqueous solution interface and used a description of the diffuse double layer based on the classical Gouy-Chapman theory [31,34]. 

The theory of the space-charge region in semiconductors was developed by Shottky [87, 88], Mott [89], Davidov [90, 91], Brattain [92] and several other authors. The idea concerning the effect of adsorption on characteristics of SCR of the semiconductor adsorbent was proposed by Hauffe [5, 6]. This theory was developed further by Volkenshtein and his group [4, 93, 94] as well as by Mark [95, 96], Morrison [21] and other scientists. Let us briefly dwell on general features of the model. 

The basic difference between metal-electrolyte and semiconductor-electrolyte interfaces lies primarily in the fact that the concentration of charge carriers is very low in semiconductors (see Section 2.4.1). For this reason and also because the permittivity of a semiconductor is limited, the semiconductor part of the electrical double layer at the semiconductor-electrolyte interface has a marked diffuse character with Debye lengths of the order of 10 4-10 6cm. This layer is termed the space charge region in solid-state physics. 

For a sufficiently large potential increase, the charge in the interphase finally corresponds to the minority charge carriers (Fig. 4.12D). The greater the width of the forbidden band eg, the broader is the potential range A f in which the space charge region has the character of a depletion layer, i.e. is formed by ionized impurity atoms. 

The presence of surface states results in a certain compression of the space charge region and leads to a decrease in the band bending. 

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