Depletion layers

As we have discussed earlier in the context of surfaces and interfaces, the breaking of the inversion synnnetry strongly alters the SFIG from a centrosynnnetric medium. Surfaces and interfaces are not the only means of breaking the inversion synnnetry of a centrosynnnetric material. Another important perturbation is diat induced by (static) electric fields. Such electric fields may be applied externally or may arise internally from a depletion layer at the interface of a semiconductor or from a double-charge layer at the interface of a liquid. 

This expression is the sum of a transient tenu and a steady-state tenu, where r is the radius of the sphere. At short times after the application of the potential step, the transient tenu dominates over the steady-state tenu, and the electrode is analogous to a plane, as the depletion layer is thin compared with the disc radius, and the current varies widi time according to the Cottrell equation. At long times, the transient cunent will decrease to a negligible value, the depletion layer is comparable to the electrode radius, spherical difhision controls the transport of reactant, and the cunent density reaches a steady-state value. At times intenuediate to the limiting conditions of Cottrell behaviour or diffusion control, both transient and steady-state tenus need to be considered and thus the fiill expression must be used. Flowever, many experiments involving microelectrodes are designed such that one of the simpler cunent expressions is valid. 

Figure Bl.28.10. Schematic representation of an illuminated (a) n-type and (b) p-type semiconductor in the presence of a depletion layer fonned at the semiconductor-electrolyte interface.

Figure C2.15.8. The p-njimction (a) p-type andn-type materials, (b) depletion layer fonnation at the p-n interface or junction and (c) p-n junction laser action.

Lp = D r ) is the minority carrier diffusion length for electrons in the -region, (0) is the minority carrier concentration at the boundary between the depletion layer and the neutral region. The sign of this equation indicates that electron injection into the -region results in a positive current flow from p to n a.s shown in Figure 7. 

The deposition of ions at the cathode creates a depletion layer across which the ions must migrate in order to deposit. This layer can vary in thickness according to surface morphology. The depletion layer is more or less defined as the region where the ion concentration differs from that of the bulk solution by >1%. The layer thickness can be decreased by agitation. 

Figure 14-7. A MISFET in operation, (a) VK>V l/j=0 an n-lypc channel of constant thickness forms at the insulator-semiconductor interlace, (b) V, > V , Vlt - Vy, the channel is pinched ofl at the drain contact. The white area that separates the p-lype substrate from the ii-lypc contacts and channel represents the depletion layer.

Figure 14-9. Schematic view of normally on (a) and normally off (b) MESFETs at zero gate voltage. In (a) a conducting channel already exists, while in (b) the depletion layer extends all over the channel.

The principle of the depletion regime is quite similar to that occurring in MES-FETs, with the difference that, unlike the MESFET, the TFT is an insulated gate device [15]. Accordingly, Eq. (14.36), which gives the width of the depletion layer, changes to... 

Saturation turns on when the charge at drain vanishes, that is when Q(L) = 0. The saturation current can be estimated by following a method introduced by Brown and coworkers [I6 and developed further by Horowitz et al. [I7J. We assume that the accumulation layer extends from the source up to a point where V(x) — VK (sec Fig. 14-10), beyond which it turns to a depletion layer. The drain current is hence given by the sum of two integrals. 

In a MESFET, a Schottky gate contact is used to modulate the source-drain current. As shown in Figure 14-6b, in an //-channel MESFET, two n+ source and drain regions are connected to an //-type channel. The width of the depletion layer, and hence that of the channel, is modulated by the voltage applied to the Schottky gate. In a normally off device (Fig. 14-9 a), the channel is totally depleted at zero gate bias, whereas it is only partially depleted in a normally on device (Fig. 14-9 b). 

Because there is no depletion layer between the substrate and the conducting channel, the equations of the current-voltage curves are in fact simpler in the TFT than in the MISFET, provided the mobility can still be assumed constant (which is not actually the case in most devices, as will be seen below). Under such circumstances, the charge induced in the channel is given, in the case of an /l-channel, by Eq. (14.23). In the accumulation regime, the surface potential Vs(x) is the sum of two contributions (i) the ohmic drop in the accumulation layer, and (ii) a term V(x) that accounts for the drain bias. The first term can be estimated from Eqs. (14.15), (14.16) and (14.19). In the accumulation regime, and provided Vx>kT/q, the exponential term prevails in Eq. (14.16), so that Eq. (14.15) reduces to... 

Otherwise, the effect of electrode potential and kinetic parameters as contained in the relevant expression for the PMC signal (21), which controls the lifetime of PMC transients (40), may lead to an erroneous interpretation of kinetic mechanisms. The fact that lifetime measurements of PMC transients largely match the pattern of PMC-potential curves, showing peaks in accumulation and depletion of the semiconductor electrode and a minimum at the flatband potential [Figs. 13, 16-18, 34, and 36(b)], demonstrates that kinetic constants are accessible via PMC transient measurements, as indicated by the simplified relation (40) derived for the depletion layer of an n-type electrode. 

A plausible mechanism for the erosion of devices that contain Mg(OH)2 is shown in Fig. 14 (2). According to this mechanism, the base stabilizes the interior of the device and erosion can only occur in the surface layers where the base has been eluted or neutralized. This is believed to occur by water intrusion into the matrix and diffusion of the slightly water-soluble basic excipient out of the device where it is neutralized by the external buffer. Polymer erosion then occurs in the base-depleted layer. 

More subtle effects of the dielectric constant and the applied bias can be found in the case of semiconductors and low-dimensionality systems, such as quantum wires and dots. For example, band bending due to the applied electric field can give rise to accumulation and depletion layers that change locally the electrostatic force. This force spectroscopy character has been shown by Gekhtman et al. in the case of Bi wires [38]. 

We should point out that up to now we have considered only polycrystals characterized by an a priori surface area depleted in principal charge carriers. For instance, chemisorption of acceptor particles which is accompanied by transition-free electrons from conductivity band to adsorption induced SS is described in this case in terms of the theory of depleted layer [31]. This model is applicable fairly well to describe properties of zinc oxide which is oxidized in air and is characterized by the content of surface adjacent layers which is close to the stoichiometric one [30]. 

In our view the final verification was given to this conclusion in paper [66] in which simultaneous O2 adsorption on partially reduced ZnO and resultant change in electric conductivity was studied. It was established in this paper that the energies of activation of chemisorption and that of the change of electric conductivity fully coincide. The latter is plausible only in case when localization of free electron on SS is not linked with penetration through the surface energy barrier which is inherent to the model of the surface-adjacent depleted layer. 

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