Space charge width

Fig. 3. Like a photoelectrochemical cell, such a powder includes sites for photo-induced oxidation and reduction, but no external current flow accompanies these transformations. Photoactivity is also maintained as the size of the particle decreases to the colloidal range although the absorption characteristics, the quantum efficiency of charge separation, and the kinetics of interfacial electron transfer may be influenced by the particle size. On sufficiently small particles, for example, the calculated space-charge width necessary for effective band bending may exceed the dimensions of the particle.

This is an interesting result. We cannot always neglect the space-charge width compared to the recombination length internal reactions in crystals with varying disorder types will be further discussed in Chapter 9. 

The width of the relaxation zone R, which is the thickness of the kinetic interface , may differ considerably from other lengths characterizing other properties of an interface (e.g., space charge width, elastic deformation width). 

In the r.h.s. part of the figure the excess conductance is normalized with respect to the space charge width (2A ifL > 4A and L if L < 4A) being a measure of the mean conductivity of the space charge zone. This value remains constant in the regime of trivial size effects but increases according to g(L) in the regime of true size effects. 

The most significant relaxation process in terms of understanding the reaction branching ratios is the relaxation of the field accelerated minority carrier. The field further acts to localise the minority carrier in the surface region. For 1 eV of band bending and a space-charge width of 300 A (Ad lO cm ), the 1/e point in the minority-carrier distribution is localised to within 10 A of the surface. Thus the surface is expected to play a significant role in the minority-carrier relaxation. 

Figure 6.16 Junction geometry, space-charge profile and potential profile for an extended junction solar cell, (a) Planar junction (b) Mildly curved junction with curvature radius larger than the space-charge width (c) Curvature radius shghtly smaller than the space-charge width (d) Curvature radius much smaller than the space-charge width.

The second difference, the incompatibility of the surface space-charge width and, is resolved by changing the model (Moore, 1983). We assume the sample to be one-dimensional and semi-infinite, divided into the space-charge region 0[Pg.245]

One other example will suffice. By fitting experimental / versus 1/a data on a given sample to Eq. (7) at low and high bias light, it is possible to extract both Ld and the space-charge width at low light. This can be done rapidly by a computer fit. As already mentioned, for a single type of recombination center we expect Lq In the ordinary Schottky-barrier formula the... 

This hypothetical visualization of charge distribution near the interface (i.e. its presence in one plane) reveals the possibihty of formation of charges in two parallel planes. One plane appears at the interface (which is populated with the majority carriers at X = 0) and another at a depth w (which is populated with charges equivalent to total charges developed at x = —w). The depth between these two planes, (Fig. 4c), is called the space charge width designated as w (for -type) and Wp (for p-type). It should be understood that concentration of minority carriers at x = — w would be much greater than the intrinsic concentration of hole at X —a. In this mathematical model, it is assumed that... 

Equation (21) can be slightly modified to include a term quantum yield (some useful parameters of semiconductors, for example, band gap, diffusion length, space charge width, and so on. Quantum yield is defined as the ratio of photocurrent (7photo) to the total flux of light (/°) used to illuminate the semiconductor that is, quantum yield... 

In Eq. (22), there are two variables, a (absorption coefficient of the semiconductor), which depends on the wavelength of light, and w, (the space charge width), which is related to the dopant s concentration and magnitude of the bias. Therefore, it is... 

Under certain conditions it may still be possible to determine the donor density of a porous photoelectrode from a Mott-Schottky measurement. This is the case for high donor densities, when the space charge width is small and is therefore stiU able to track the surface contours. An example of this for nanostructured Si-doped Qt-Fe203 photoanodes is shown in Fig. 3.18, reported by Cesar et al. [57], The actual surface area was estimated by dye absorption experiments, and the estimated donor density for this system was 10 ° cm . The concave shape of the Mott-Schottky plot is consistent with a gradual decrease in the effective surface area (cf. (3.5)) as the potential is increased and the depletion layer progressively penetrates into the bulk [58]. 

Lifetimes in hematite are much shorter than Ti02. Carrier relaxation to the conduction or valence band edge occurs within a few hundred fs, and recombination or trapping occurs within 3-5 ps. To have Tt = tr, the doping density should be 10 °, or approximately 0.25 at.%, assuming that these values for lifetime are similar to Tr. This corresponds to a space charge width of 5 mn for a potential drop of 0.25 V. Therefore, it would be desirable to synthesize highly doped hematite nanostructures with dimensions of less than 10 nm. 

Two distinct differences can be seen in the relation for the space charge width in the Mott-Schottky compared to the Gouy-Chapman boundary conditions. When the majority defect carmot redistribute, the space charge width is dependent on the space charge potential, and the depletion width is greater in spatial extent due to a reduced charge screening ability. 

Fig. 2 Comparison of characteristic macropore sizes on p-Si in the current-line-driven regime, when changing current density for a substrate resistivity of 100 Q cm (a) and silicon doping for an applied current density of 10 rtiA/cm (b) (After Chazalviel et al. 2002). Triangles refer to the wall width and diamonds to the pore diameter the closed (open) symbols refer to the data obtained in 35 % (25 %) ethanolic HR The solid lines refer to the theoretical prediction (Chazalviel et al. 2002) for the pore diameter, and the dotted line is two times the space-charge width X...

The question still arises, however, concerning which rate constant is expected in order to fulfill the condition that the hot electron is transferred before it relaxes at the bottom of the conduction band. In this context, it should be mentioned that there may be quantization effects at the surface of highly doped III-V semiconductors because of the small space charge width (100 A) the relaxation time may be longer on the surface compared with the bulk. Assuming this effect to be small, the relaxation time is of the order of 10 ps (see above). Then hot electron transfer can only occur if the time of the electron transfer from the semiconductor surface to an acceptor in the solution is comparable with this relaxation time. 

The space-charge width, w, which is related to the majority carrier density, N , and the potential drop, space charge region by the equation ... 

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