Potential drop in the substrate

For a solid between two spherical surfaces shown in Fig. 8.65, the resistance can be described by 

FIGURE 8.65. The change of voltage due to resistance of the material in the substrate of a hollow sphere as a function of radius of curvature of the inner sphere. 

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For a moderately or highly doped material the potential drop due to ohmic resistance in the substrate is very small. For example, for a substrate with a resistivity of 0.1 Qcm ( 1017/cm3) the potential drop in the substrate of 0.1 mm thick at a current density of 10... 

Oxide formation on Si follows the high field law, but because of the influence of the potential drop in the substrate it will be discussed in Sect. 3.2.3.3.3. 

Physically, the sensitivity of reactions to surface curvature can be associated with the space change layer or the resistance of the substrate. For moderately or highly doped materials, this sensitivity is only associated with the space change layer because the ohmic potential drop in the semiconductor substrate is very small. However, for lowly doped material a significant amount of potential can drop in the semiconductor to cause the current flow inside semiconductor to be also sensitive to the curvature of the surface. 

Figure 26. Potential drops along the current path in a pore AVsi = potential drop in silicon substrate, AVS = potential drop in the space change layer AV0I = potential drop in oxide AVh = potential drop in the Helmholtz layer, AVei = potential drop in electrolyte.

When the resistance of the substrate is high and a significant amount of potential is dropped in the substrate, the potential drop may not be uniform along a curved pore bottom due to the nonlinear potential distribution on the material surrounding the bottom. Formation of macro PS on lowly doped materials becomes possible under such a condition. 

Due to ambient humidity, one or very few layers of absorbed water formed due to condensation results in the formation of water meniscus between the conductive tip and the substrate. Thus highly resistive electrolyte is formed by a thin water film in wet gas atmosphere which facilitates formation of nanostructures by oxidation of the metal substrate. The introduction of reference electrode in the electrochemical nanocell is not at all possible due to space constraint. A large potential drop in the electrolyte and at the counter electrode, i.e., AFM tip is encountered due to the absence of reference electrode. Several attempts have been made for nanostructure formation on metal substrates such as Si,... 

Electrical field effects are an example of a transport phenomenon that does not arise in most chemical reactors, and these field effects often dictate the current distribution. Usually, electrical field effects are more important in the (ionicaUy conducting) electrolyte than in the (electronically conducting) electrodes. However, as is the case of porous electrodes for fuel cells and batteries, significant potential variations in the electrodes may result if the electrodes are very thin, very large, or have high specific resistivity. Current distributions where the potential drop in the electrode is important were first studied in 1953 [4] the phenomenon is called the terminal effect or resistive substrate effect. ... 

Because there is no depletion layer between the substrate and the conducting channel, the equations of the current-voltage curves are in fact simpler in the TFT than in the MISFET, provided the mobility can still be assumed constant (which is not actually the case in most devices, as will be seen below). Under such circumstances, the charge induced in the channel is given, in the case of an /l-channel, by Eq. (14.23). In the accumulation regime, the surface potential Vs(x) is the sum of two contributions (i) the ohmic drop in the accumulation layer, and (ii) a term V(x) that accounts for the drain bias. The first term can be estimated from Eqs. (14.15), (14.16) and (14.19). In the accumulation regime, and provided Vx>kT/q, the exponential term prevails in Eq. (14.16), so that Eq. (14.15) reduces to... 

In most systems the substrate electrodes are larger than the powered electrodes. This asymmetric configuration results in a negative dc self-bias voltage Vdc on the powered electrode. Without that, the difference in electrode areas would result in a net electron current per RF period [134, 169]. It has been shown that the ratio of the time-averaged potential drops for the sheaths at the grounded (V g) and the powered electrode (Vsp) are inversely proportional to a power of the ratio of the areas of the two electrodes (Ag, Ap) [134, 170-172] ... 

There are five possible physical phases in the current path in which the current conduction mechanisms are different as illustrated in Figure 19. They are substrate, space charge layer, Helmholtz layer, surface oxide film, and electrolyte. The overall change in the applied potential due to a change of current density in the current path is the sum of the potential drops in these phases ... 

For moderately doped substrates, when the surface is free of oxide the change of potential is mostly dropped in the space charge layer and in the Helmholtz double layer. The reactions are very sensitive to geometric factors. The reaction that is kinetically limited by the processes in the space charge layer is sensitive to radius of curvature, while that limited by the processes in the Helmholtz layer is sensitive to the orientation of the surface. Depending on the relative effect of each layer the curvature effect versus anisotropic effect can vary. 

We then study experimentally the effect of an inert electrolyte solution and show that ion motion forces an applied electrical potential in the dark to drop near the substrate electrode, thus reinforcing the effects of the distributed resistance. Overall, the 2 conduction and valence bands (whose spatial gradients reflect the electric field) remain approximately flat both at equilibrium and under illumination therefore, charge transfer occurs primarily by diffusion rather than by field-induced drift [4,40-42]. Recent numerical simulations [43,44] and modeling of photogenerated trapped charges [45] show that in an illuminated DSSC there may be, in fact, a very small bulk electric field of about 0.1-3 mV/pm, but this is not expected to have much influence. 

In these experiments, the potential distribution was measured under conditions where the interfacial current density was minimized by the use of an inert electrolyte. If the electron-transfer rate across the interface had truly been zero (Ret = °°), the whole 2 film would have eventually charged up to the applied potential it was the unavoidable leakage current across the interface and the relatively short time scale of our experiments that prevented this from happening. These experiments show that even when Rct is maximized, ion motion through the nanoporous film causes the applied potential to drop near the substrate electrode in nonilluminated DSSCs. As we showed earlier, decreasing Rct causes the applied potential to drop even closer to the substrate electrode. 

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