Depletion layer, recombination

Mechanisms 1 and 2 are included in the model that is used here for comparison with experimental data. Interface recombination and dark current effects are not included however, the experimental data have been adjusted to exclude the effects of dark current. To include the additional bulk and depletion layer recombination losses, the diffusion equation for minority carriers is solved using boundary conditions relevant to the S-E junction (i.e., the photocurrent is linearly related to the concentration of minority carriers at the interface). Using this boundary condition and assuming quasi-equilibrium conditions (flat quasi-Fermi levels) ( 4 ) in the depletion region, the following current-voltage relationship is obtained. 

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... 

We now consider the form of the expression for the collection efficiency in eqn. (426). It consists of a numerator 1 - e"3tv + (aLp/[aLp + l])e ,v = Xgm, a generating term, and a denominator that contains a depletion layer recombination term and a diffusion loss term. The last term is likely to be negligible at the higher potentials considered here and we ignore it. The recombination term in the denominator has the general form K/k zpB and... 

If we assume that internal losses through depletion-layer recombination are small, we may write N = Xgen if, on the other hand, depletion layer recombination must be taken into account and N Xgen, then N k zF, where F is defined, from above, as... 

Case (1). Depletion-layer recombination significant. N k zF and ps F 0 whence the fractional occupancy ft is given by Kn ... 

Fig. 100. Equivalent circuit for a.c. modulation under illumination when depletion-layer recombination cannot be neglected.

Photocurrent voltage curves have been studied with molybdenum selenide crystals of different orientation and different pretreatment. Figure 5 represents results for three typical surfaces of n-type MoSe (JJ+). An electrode with a very smooth surface cleaved parallel to the van der Waals-plane shows a very low dark current in contact with the KI containing electrolyte since iodide cannot directly inject electrons into the conduction band and can only be oxidized by holes. At a bias positive from the flat band potential U where a depletion layer is formed a photocurrent can be observed as shown in this Figure. This photocurrent reaches a saturation at a potential about 300 mV more positive than when surface recombination becomes negligible. 

It thus appears that even on metal-free SrTi03 conduction-band electrons are the primary reductants. Since similar reaction rates occur on pre-reduced and stoichiometric crystals with disparate depletion layer widths, the electrons do not tunnel through the depletion layer. With no Pt to provide an outlet for electrons at potentials far positive of the flatband potential, strong illumination would flatten the bands almost completely and allow electrons to reach the semiconductor surface. The presence of both electrons and holes at the surface could lead to unique chemistry as well as high surface recombination rates. 

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. 

If the only process occurring at the surface of p-type material, at x = — W, is recombination and no faradaic process takes place, then the total current flowing out of the depletion layer, - D [d(Sn)/dx]x n together with the recombination rate stdnw, must balance the total generation rate, provided no recombination occurs within the depletion layer. From eqn. (381)... 

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. 

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. 

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. 

Examination of Fig. 69 shows that, within the depletion layer, p first falls and then, near the surface, rises rapidly. It follows from eqn. (435) that the recombination law is likely to remain first order near the depletion layer boundary and, if there is a change of mechanism, it will occur well within the depletion layer. This is an important simplification in two respects. 

The recombination rate will only be significant near the surface, for a value of x < xRZ where xnz will be calculated below. For the various possible transitions from recombination law I near the inner depletion-layer boundary to either II or III near the surface, we have... 

The most important case that we need to consider here is the I - III transition. In this case, above some critical bias potential u8, there exists a point in the depletion region, xRZ, at which the dominant recombination mode changes from first order in p to first order in n (and zeroth order in p). The total recombination flux within the depletion layer is... 

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. 

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