Mott-Schottky approximation

Within the Mott-Schottky approximation, qsc = (2e0 ace0ND)1/2 [large values of eacND, (pnl4>sc can become significant and some examples are shown in Fig. 27 [67] for a value of sc appropriate to Ti02 ( 173). It can be seen that cpn is an appreciable fraction of the total potential, especially near flat-band potential. From the formula above, the equivalent circuit is... 

The complexity afforded by these traps arises from the fact that the frequency range normally used in a.c. studies, 1-105 Hz, is likely to contain the relaxation frequency appropriate to such trap sites. To explore this concept, we may consider Fig. 32 [76]. We identify a deep trap level of energy E0 relative to the conduction band, lying below the Fermi level in the bulk of the semicondutor and therefore ionised only up to a distance l into the depletion layer. For x > l, only the shallow traps are ionised at equilibrium. Within the spirit of the Mott-Schottky approximation, we obtain (provided V 3= V0)... 

J. F. DEWALD (Bell Telephone Laboratories) I do not believe that the effect of semiconductivity on catalysis is completely contained in either the Schottky or the Mott-Schottky approximations. Consider a supported metal catalyst. If one has an exhaustion layer on an n-type support then the total charge in the exhaustion layer would be very small. The potential, acting over so large a distance, would have very little effect on catalysis. If, however, you can cause an electron enrichment layer in the vicinity of the metal-n-type contact, you might expect very much larger catalytic effects, for there would be much larger electric fields near the point of contact. 

Finally, within the Mott-Schottky approximation (Eq. 11), large values of or Ad can lead to the ratio Fh/ Fjc becoming significant. Figure 11 contains estimates of this ratio for several values of Aq for a semiconductor with a large s value (Ti02, s = 173) mapped as a function of the total potential drop across the interface [50]. Clearly, Fh can become a sizeable fraction of the total potential drop (approaching the situation for metals) under certain conditions. It has been shown [51] that the Mott-Schottky plots will still be linear but the intercept on the potential axis is shifted from the Fn, value. 

Equation (13) represents an ordinary differential equation requiring two boundary conditions and a reference point for the potential, commonly set to zero in the bulk. If ionic defects are frozen in place and unable to move, the charge density in Poisson s equation is determined only by redistribution of electronic charge carriers that are ionized from the dopants. This is known as the Mott-Schottky approximation [18], which results in a simplification to Poisson s equation by setting Cj(x) = Cj ... 

Thus, we have to conclude that, without knowing the physical nature of the frequency dependence of the differential capacitance of a semiconductor electrode, the donor (or acceptor) concentration in the electrode cannot be reliably determined on the basis of the Schottky theory, irrespective of the Mott-Schottky plot presentation format. Therefore, the reported in literature acceptor concentrations in diamond, determined by the Schottky theory disregarding the frequency effect under discussion, must be taken as an approximation only. However, we believe that the o 2 vs. E plot (the more so, when the exponent a approaches 1), or the Ccaic 2 vs. E plot, are more convenient for a qualitative comparison of electrodes made of the same semiconductor material. 

Normally, for semiconductors, Csc < CH so CT Csc. Roat may be varied systematically and the decay of j can often be approximated by a single exponential form, i.e. kr 1/RtCi. or kT potentials well positive of V, the long-time transient time-constant t (Rm + Rout)Csc, and a plot of x vs. R]oad (sflin + Rout) is linear, as shown in Fig. 105. Confirmation of this is obtained from the fact that 1/t2 obeys the Mott-Schottky relationship. At potentials close to V, kec becomes much larger and the decay law more complex. 

For various illumination intensities, the diameter of the semicircle fitting the data at high frequencies equals approximately kT/ely pHl [45-47, 49]. In addition, it was shown that upon illumination, a capacitive peak appears in the C versus V plot of the n-GaAsjO.l M H2SO4 interface [45,46, 51], The peak value proved to be a function of the frequency and the photocurrent density as measured in region G [51]. This behavior is markedly different from the purely capacitive impedance (vertical line in the Nyquist plane and straight Mott-Schottky plot) expected for a blocking s/e interface (see Sect. 2.1.3.1). 

Approximate solutions for the Gouy-Chapman case were presented by Maier [21], while more recently analytical solutions for the Gouy-Chapman as well as the Mott-Schottky cases were presented by litzehnan et al. [22, 23]. The relevant expressions for each partial conductivity are summarized for the Gouy-Chapman case ... 

It now remains for us to calculate the resistance of a Mott-Schottky boundary layer, that is a depleted boundary layer in the extrinsic case with nonmobile dopant. As we saw for very severe depletion, the total effect took the form of an almost stepwise function at A ( A - V ). In this extreme step-function approximation it follows (AZ Z ) that... 

In the case of the Mott-Schottky boundary layer (majority charge carrier immobile, counterdefect depleted), we obtained a simple result for high depletion and could ap-proximate the space charge profile by means of a rectangular function of width A oc Ay l o total surface charge is approximately obtained by multiplication of A with the constant doping concentration m resulting in ... 

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