Recombination in the depletion layer

Recombination in the depletion layer can become important when the concentration of minority carriers at the interface exceeds the majority carrier concentration. Under illumination minority carrier buildup at the semiconductor-electrolyte interface can occur due to slow charge transfer. Thus surface inversion may occur and recombination in the depletion region can become the dominant mechanism accounting for loss in photocurrent. 

The most inaccurate assumption, especially at low band-bending, is undoubtedly the first. However, the mathematical derivation leading to eqns. (396) and (397) cannot be easily modified to take account of depletion-layer recombination owing to the way in which the depletion layer is considered. In order to develop the theory to take into account recombination in the depletion layer, it is necessary to solve explicitly the transport equation in the depletion layer as well as the bulk. If we persevere, for the moment, with the Schottky barrier model and we continue, for the moment, with the assumption that recombination does not occur in the depletion layer (x < VF) then the transport equations for x < VF, x > VF are... 

The problem of recombination in the depletion layer is not at all trivial to solve. The only analytical solution that has been obtained as yet [142] is that for first-order recombination. An examination of eqn. (352) shows that this will only hold ifp < n or, equivalently, Ap NDev% a result only likely... 

The significance of the various terms in eqn. (426) are shown in Fig. 68. The only difference between this and the formula derived by Wilson, eqn. (415), is the presence of an additional loss term in the denominator which reflects recombination in the depletion layer. However, unlike the loss term representing diffusion out of the depletion layer into the bulk, the depletion recombination term decreases only slowly with increasing bias and plays a significant role even at comparatively large vs values. However, it is clear that, at comparatively large values of y (ca. 0.1) and reasonable values of the ratio (Dplk z)LD, very low efficiencies are found, even at high bias potentials. This is quite unreasonable, as will be seen below. 

Fig. 99. Equivalent circuit for a.c. modulation under illumination when recombination in the depletion layer can be neglected.

Experimental studies [188] show that in the case of n-GaAs eleetrodes in contact with Ce + as the hole injection agent, surface recombination prevails. On the other hand, with n-GaP electrodes, recombination in the depletion layer must also be... 

There is no recombination in the depletion layer. That is, all the holes optically generated in the bulk and within the depletion layer (Fig. 18) are swept to the surface without loss ... 

The structure of eqn. (415) is very revealing the numerator consists of two generating terms. One, (1 — e a,r), is the generation of holes in the depletion layer and the second, (aLp/[ocLp + l])e a,v, the generation of holes in the bulk modulated by a recombination factor. The second term in the denominator represents the flux of carriers out of the depletion layer into the bulk it therefore represents a loss of efficiency. 

Equation (423) represents the best that has been achieved hitherto using formal analytical procedures and the problem, as has already been emphasised, lies with the nature of the recombination formula, eqn. (352). As the band-bending increases, n must decrease to the point where a shift in the kinetic law is expected at some point in the depletion layer. 

More complex treatments have been proposed, in which efforts have been made to separate hole and electron currents in the depletion layer [7]. The circuit of Fig. 99 illustrates the situation for a semiconductor under depletion or inversion conditions in which zero recombination occurs in the depletion layer [175]. In the figure, the suffix n refers to electrons and Q is a capacitance associated with inversion (if this is operative). The impedance Zr describes the generation of holes and their recombination in the bulk of the semiconductor. 

If bulk recombination is important in the depletion layer, then we cannot separate hole and electron flows in the above manner and the Zr, / scp network collapses to a frequency-independent resistor I D, as shown in Fig. 100. In this figure IFis a Warburg impedance for the hole current. This is too complex, as it stands, for analysis and a simpler case can be derived if Css is dominant and the frequency range is such that W can also be neglected. Under these circumstances, I D, Raan and 7 ssp further collapse to a simple resistor Rr, leading to the equivalent circuit shown in Fig. 101, which has been applied to p-GaAs under illumination and n-GaAs under hole injection. 

The optical absorption arising from the defect transitions is weak because of the low defect densities and in a thin film cannot be measured by optical transmission. The techniques of PDS, CPM and photoemission yield, described in Section 3.3, have sufficient sensitivity. Photocapacitance, which measures the light-induced change in the depletion layer capacitance, is similarly sensitive to weak absorption (Johnson and Biegelsen 1985). PDS measures the heat absorbed in the sample and detects all of the possible optical transitions. At room temperature virtually all the recombination is non-radiative and generates heat by phonon emission. CPM detects photocarriers and so is primarily sensitive to the optical transitions which excite electrons to... 

Harima et al. (1989) measured the photogeneration efficiencies of 5,10,15,20-tetraphenylporphyrin (TPP) and its Zn complex (ZnTPP) doped with tetracyano-quinodimethane (TCNQ), o-chloranil (Chi), phenothiazine (Pz), and I. TCNQ, Chi, and I are electron acceptors while Pz is a donor. The porphyrins were selected on the basis of differences in exciton diffusion lengths (Tanimura et al., 1980 Yamashita et al., 1987) and oxidation potentials (Felton, 1978). The photogeneration of I doped ZnTPP was described by a direct ionization process via a singlet state of ZnTPP and ascribed to a reduction of the electron-hole recombination rate in the depletion layer. For TPP doped with Chi, Pz, or I, the results were explained by exciplex dissociation (Loutfy and Menzel, 1980). For... 

In single-crystal electrodes the electric field in the depletion layer of the semiconductor separates the charges and decreases the probability of charge-pair recombination [30]. However, in small-particle colloids such a depletion layer does not exist due to the nanometer particle size, and there is no electrical field to separate the charges [36]. Due to the large recombination rate in small-particle colloids, the lifetime of charged pairs is very short, and only very fast reactions with adsorbed species can lead to efficient charge separation. In order to facilitate chemical pro-... 

With the assumptions that (1) there is negligible recombination in the space charge layer and at the surface and all the carriers generated in the space charge layer are driven by the field to the surface, and (2) the electrode reactions are sufficiently fast, using Eq. (1.83) the contribution due to photogeneration in the depletion layer can be described by... 

This high photovoltage is remarkable because it is close to the thermodynamically possible value of around 0.7 V (see also Section 11.1.1.3). Such a high value has not even been achieved in simple p-n homojunctions, for which the lower photovoltage has been interpreted as recombination losses in the depletion layer [23]. None of these losses seem to occur in the n-Si/CH OH liquid junction. Residual losses in the short-circuit arise from optical reflectivity and absorption processes and losses in the fill factor arise from concentration overpotentials and uncompensated series resistance losses from the potentiostat [20]. Neglecting the latter losses, this cell is the first system with which the same efficiency was obtained as that found with Si homojunctions. 

Fig. 5. An n-type semiconductor electrode under depletion conditions, illuminated from the electrolyte side. Holes generated in the depletion layer and in the diffusion layer (width L) reach the surface and can contribute to the photocurrent. Holes generated deeper than dsc + L are lost by bulk recombination.

W. J. Albery and P. N. Bartlett, The recombination of photogenerated minority carriers in the depletion layer of semiconductor electrodes, J. Electrochem. Soc. 130 (1983) 1699-1706. 

---