Capacity space-charge region

Samec et al. [15] used the AC polarographic method to study the potential dependence of the differential capacity of the ideally polarized water-nitrobenzene interface at various concentrations of the aqueous (LiCl) and the organic solvent (tetrabutylammonium tetra-phenylborate) electrolytes. The capacity showed a single minimum at an interfacial potential difference, which is close to that for the electrocapillary maximum. The experimental capacity was found to agree well with the capacity calculated from Eq. (28) for 1 /C,- = 0 and for the capacities of the space charge regions calculated using the GC theory,... 

The Differential Capacity Due to the Space Charge. When capacity measurements are carried out on a semiconductoi/electrolyte interface, one must not forget that the space-charge region inside the semiconductor has the ability to store charge. The contribution of this region to the differential capacity of the interface can... 

Here Lse is the thickness of the space-charge region in a semiconductor related to the capacity Csc by Csc = E0EaJLsc. In particular, for the depletion layer we have, according to Eq. (19),... 

The capacity found to be frequency-independent, is assumed to reflect the voltage drop in the space charge region, and the inability to reach a low capacity simply means that the voltage drop appears across the oxide. 

Due to the great extension of the space-charge region, almost all the potential drop occurs across it. So we can measure its capacity, Csc, and calculate from the Mott-Schottky relation... 

Without going into much mathematical detail, the capacity of the space charge region is derived by solving the Poisson-Boltzmann equation as... 

The capacity of the space-charge region czm be related to the dopant concentration (or fixed charge) in a semiconductor. The space-charge region is essentially equivalent to the diffuse double layer treated in electrolytes with the exception that ionized impurities are present that, at room temperatures, are immobile. For this case, Poisson s equation becomes... 

The capacity of the space-charge region in a semiconductor Csc, under the formation of a depletion layer, is related to the potential drop in this region by ... 

The capacity measured is assumed to represent only the capacity of the space-charge region in the semiconductor and not to include, for example, the capacity of surface states, adsorption capacity, etc. In certain cases, this condition is satisfied, for example, on a zinc oxide electrode but more frequent is the situation where the contribution of the capacity of surface states is considerable. 

In other words, it is supposed that (1) the electrode capacity measured is entirely determined by the capacity of the space-charge region in the semiconductor and (2) a change in the electrode potential leads only to a change in the potential drop in the semiconductor, while the potential drop across the Helmholtz layer remains constant. 

An explanation of this fact was offered by Pereira et al. [11] and by Huber et al. [12] the overlap of the two solvents at the interface entails an overlap of the two space-charge regions. Thus the average separation between the opposing charges is reduced, and the capacity enhanced. Both groups performed explicit calculations the former employed the quasichemical approximation, while the latter performed Monte Carlo simulations. To some extent these two techniques complement each other the quasichemical approximation should be good at low electrolyte concentrations, while the Monte Carlo simulations, because of the finite ensemble size, are performed for more concentrated solutions. 

Thus, if the semiconductor corresponds essentially to an insulator of the Schottky barrier type, use of Mott-Schottky plots will allow the determination of the capacitance of the inner layer. Utilization of impedance measurements with different frequencies may give rise to the possibility of determining the double layer capacity separate from the inner layer. In this way a map of the double layer, an estimation of the Helmholtz potential difference and the potential difference (pd) in the space charge region may be obtained. The pd in the Helmholtz layer is, however, not only given by the charge on the surface of the polymer, but also by the potential difference due to aggregated layers which form within it, and in particular the solvent dipole layer (70). 

Consider a plane metal electrode situated at z = 0, with the metal occupying the half-space z < 0, the solution the region z > 0. In a simple model the excess surface charge density a in the metal is balanced by a space charge density p(z) in the solution, which takes the form p(z) = Aexp(—kz), where k depends on the properties of the solution. Determine the constant A from the charge balance condition. Calculate the interfacial capacity assuming that k is independent of a. 

It will be seen that the values of the space-charge capacities are low (-0.01-1 fiF cm 2) compared with the capacities (-17 (J.F cm 2) of the region between the semiconductor surface and the OHP plane, the Helmholtz-Perrin parallel-plate region. That is why the space-charge capacities (the inverted parabolas) are noticed, for the observed capacity is given by two capacitors in series, the space charge, Csc, and Helmholtz-Perrin HP capacitors. Thus,... 

Assuming a roughness factor of 2, the theoretical values of the capacities for the doped samples were also nearly obtained at the highest frequencies used. Stronger deviations occurred when the minority carriers were enriched in the space charge, i. e., for the n-type samples in the more anodic, for the p-type samples in the more cathodic region. 

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