Minority Carrier Transfer Processes

In the case of minority carrier injection, the interfacial current is not only determined by the electric field but also by diffusion of the carriers. The hole current 

In principle, the same derivation can be applied as that used in Section 2.2.3. The only difference is that the minority carrier current is not only determined by the hole injection and recombination in the n-type region but also by the injection of electrons into the p-type area. Thus, we now have instead of Eq. (2.31) 

Equation (2.36) is the famous Shockley equation which is the ideal diode law [16]. According to Eq. (2.35), we have obtained again the same basic current-voltage dependence as already derived for majority and minority carrier devices with semiconductor-metal junctions (see Eqs. (2.18) and (2.31)). As already mentioned, the physical difference occurs only in the pre-exponential factor / g. The general shape of a complete j-U curve in a linear and semilog plot has already 

The factor n equals 2 when the recombination current dominates and n equals 1 when the diffusion current dominates. When both currents are comparable, n has a value between 1 and 2. 

Sometimes an ideality factor of greater than 1 is also reported for a majority carrier device. In this case, however, there is no physical basis for an ideality factor of 1 and any deviation from = 1 must have technological reasons. 

A system in which only majority carriers (electrons in n-type) carry the current, is frequently called a majority carrier device . On the other hand, if the barrier height at a semiconductor-metal junction reaches values close to the bandgap then, in principle, an electron transfer via the valence band is also possible, as illustrated in Fig. 2.8a. In this case holes are injected under forward bias which diffuse towards the bulk of the semiconductor where they recombine with electrons ( minority carrier device ). It is further assumed that the quasi-Fermi levels are constant across the space charge region i.e. the recombination within the space charge layer is negligible. In addition Boltzmann equilibrium exists so that we have according to Eqs. (1.57) and (1.58) 

As already mentioned, we have a forward bias if the n-type semiconductor is made positive with respect to the metal. Under these conditions holes are injected into the semiconductor and we have pn In this case the quasi-Fermi level of holes, Ep p, occurs below that of electrons, Ep , in Fig. 2.8, i.e. it is closer to the valence band which is equivalent to the fact that the minority carrier density is increased. The externally applied voltage is then determined by 

If there is a strong coupling between the metal and the semiconductor, Epp is close to Ep in the metal at all potentials. For a reverse bias (negative U), there is an extraction of holes and Epp occurs above Epn as shown in Fig. 2.8b. The resulting current-voltage curve can be derived as follows. 

31) is identical to Eq. (2.18) derived for a majority carrier device (thermionic emission model). Accordingly, the same type of current-voltage curve is expected as that given in Fig. 2.7. The characteristics of the models occur only in the preexponential factors, which indeed are different in both cases (compare Eqs. 2.17 and 2.30). As mentioned before the jo of the majority carrier device is only determined by the barrier height and some physical constants (Eq. 2.19), whereas the y o of the minority carriers depends on material-specific quantities such as carrier density, diffusion constant and diffusion length. 

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This relation is identical to that derived for a pure solid state device which is determined by minority carrier transfer and recombination, such as a pn junction (see Section 2.3) or semiconductor-metal contact (see Section 2.2.3). The corresponding current-potential curves in the dark and under illumination are given by the solid lines in Fig. 7.16. Taking the complete Eq. (7.71), there may be a certain potential range where the recombination current determines the process until the current levels off to a constant jy. For very large jy values, the cathodic current can ultimately be diffusion-limited, which can be checked experimentally by using a rotating electrode. 

Fig. 7.58 a) Excitation of electron hole pairs and minority carrier transfer, b) Injection of minority carriers from the solution and recombination processes... 

Also of interest in connection with Fig. 16 is a process that has been labeled hot transfer (see, for example, Toyozawa, 1978 Kayanuma and Nasu, 1978 Jortner, 1979). Here it is suggested that in the case AE > A the transition to the ground state can take place during the lattice relaxation, i.e., before the excited state has reached its equilibrium position. This effect was first suggested by Dexter et al. (1955) and further analyzed by Bartram and Stoneham (1975) for F centers. A recent slight modification, in terms of fast capture of a majority carrier subsequent to that of a minority carrier, has been suggested by Sumi (1981) he points out that this process may be active in recombination-enhanced defect reactions. 

In many PEC systems the chemical kinetics for the primary charge transfer process at the interface are not observed at the light intensities of interest for practical devices and the interface can be modeled as a Schottky barrier. This is true because the inherent overpotential, the energy difference between where minority carriers are trapped at the band edge and the location of the appropriate redox potential in the electrolyte, drives the reaction of interest. The Schottky barrier assumption breaks down near zero bias where the effects of interface states or surface recombination become more important.(13)... 

It should be mentioned further, that the shift of energy bands upon creation of minority carriers does not only occur upon light excitation. Holes can also be produced in the dark via hole transfer from a hole donor into valence band. This kind of process occurs for instance by using the oxidized species of the redox couple [Ru(bipy)3]2+/3+. A corresponding cathodic dark current starts at the same potential at which the photocurrent onset was found in the presence of the reduced form of the same couple. This shift of Efb by hole injection from the electrolyte has been found also with Ce4+ at n-WSe2 (McEvoy et al., 1985) and n-GaAs (Schroder et al., 1985). 

In the previous section, we considered electron transfer processes involving only the majority carrier band. For semiconductors with lower band-gaps, either doped or intrinsic, both bands may be involved and we consider two current contributions arising from the two bands as shown in Fig. 41. In practice, it is found that the dominant contribution usually arises from only one of the two bands if the majority carrier band is involved, the treatment will reduce to that given above, but if the minority carrier band is the more significant, then the current may be limited by the rate of thermal generation and transport of the majority carriers to the surface. 

Physically, kac and kK represent the rate constants for discharge of (a) the depletion layer charged by minority carriers that recombine through bulk or surface processes and (b) the faradaic transfer processes at the interface. 

Phenomena which involve the concept of minority carriers are perhaps most characteristic of semiconductor electrodes. The study of charge-transfer processes is one of the most direct ways of getting at the details involved. Dewald (4) has given an introduction to the subject, and in the present section we extend some of these ideas to cover the response to transient conditions. 

Most probably, the real situation in these systems is one of compromise between the two possibilities discussed above, Le., the initiator is a mixture of perdhloric acid and acetyl perchlorate. It seems obvious to us that the mechanism of these polymerisations is psoidocationic and the minor effect of added perchlorate salts is simply due to homoconjugation of perchloric acid present at the beginning of the reaction or liberated in ontaneous transfer processes, and not to common-ion effects suppressing the concentration of free ionic chain carriers. 

The back-electron-transfer step is identical to the band-to-molecular state charge-transfer process involving minority-carrier injection (Section 2.3.5). This step is sensitive to defects that would act as donors with smaller activation barriers than a single regeneration step involving band-to-molecular acceptor transitions. Optimally the surface should have low defect densities and the cation level should lie as high above the valence band as possible (electrochemical determinations of the redox potential of the cation radical are needed). 

Figure 12.14 Band diagram of the semiconductor-electrolyte interface, illustrating photogeneration and loss of minority carriers (holes) via surface recombination and interfacial electron transfer. The balance between these processes determines the steady-state surface concentration of minority carriers...

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