The Depletion Layer

There is a characteristic region within the semiconductor within which the charge would have been removed by the equilibration process. Beyond this boundary, the ionized donors (for a n-type semiconductor), have their compensating 

eo is the electronic charge and Eg is the static dielectric constant of the semiconductor. The potential distribution is mapped in Fig. 6. We are now in a position to quantify the parameter Vsc  

Mapping of the Semiconductor Band-edge Positions Reiativeto Soiution Redox Levels 

Considerations of interfacial electron transfer require knowledge of the relative positions of the participating energy levels in the two (semiconductor and solution) phases. Models for redox energy levels in solution have been exhaustively treated elsewhere [27, 28]. Besides the Fermi level of the redox system (Eq. 6), the thermal fluctuation model [27, 28] leads to a Gaussian distribution of the energy levels for the occupied (reduced species) and the empty (oxidized species) states, respectively, as illustrated in Fig. 5(a). The distribution functions for the states are given by 

As with metals, the Helmholtz layer is developed by adsorption of ions or molecules on the semiconductor surface, by oriented dipoles, or especially in the case of oxides, by the formation of surface bonds between the solid surface and species in solution. Recourse to band edge placement can be sought through differential capacitance measurements on the semiconductor-redox electrolyte interface [29]. 

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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. 

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... 

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). 

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. 

In Eq. (4.5.5), describing an n-type semiconductor strongly doped with electron donors, the first and third terms in brackets can be neglected for the depletion layer (Af0 kT/e). Thus, the Mott-Schottky equation is obtained for the depletion layer,... 

Eventually, in region (d), tunnelling occurs from the tip to the sample. Although the depletion layer is still thick, the effective thickness of the barrier in this region is actually reduced and the presence of the surface states plays a dominant role in maintaining the tunnelling current in this region. 

A constant bias potential is applied across the sensor in order to form a depletion layer at the insulator-semiconductor interface. The depth and capacitance of the depletion layer changes with the surface potential, which is a function of the ion concentration in the electrolytic solution. The variation of the capacitance is read out when the semiconductor substrate is illuminated with a modulated light and the generated photocurrent is measured by means of an external circuit. 

The working principle of LAPS resembles that of an ion-selective field effect transistor (ISFET). In both cases the ion concentration affects the surface potential and therefore the properties of the depletion layer. Many of the technologies developed for ISFETs, such as forming of ion-selective layers on the insulator surface, have been applied to LAPS without significant modification. 

The principle of depletion is illustrated in Figure 1. If a surface is in contact with a polymer solution of volume fraction , there is a depletion zone near the surface where the segment concentration is lower than in the bulk of the solution due to conformational entropy restrictions that are, for nonadsorbing polymers, not compensated by an adsorption energy. The effective thickness of the depletion layer is A. Below we will give a more precise definition for A. 

Figure 7.5 Mott-Schottky plot for the depletion layer of an n-type semiconductor the flat-band potential Eft, is at 0.2 V. The data extrapolate to Eft, + kT / eo-...

Figure 7.8 Photogeneration of holes at the depletion layer of an n-type semiconductor.

From a chemical point of view a hole at the surface of a semiconductor entails a missing electron and hence a partially broken bond. Consequently semiconductors tend to dissolve when holes accumulate at the surface. In particular this is true for enrichment layers of p-type material. At the depletion layers of n-type materials the holes required for the dissolution can also be produced by photoexcitation. 

Following Section 7.2, we consider the depletion layer of an n-type semiconductor, assuming that the concentration of holes is negligible throughout. The situation is depicted in Fig. A.l, which also defines the coordinate system employed. Starting from Eq. (7.1) ... 

Inspection of equations (33) and (45) confirms that the flux. 7, is maximal for 6 = 0. It decreases with increasing 6, and tends to zero for 6 = tt, where the depletion layer thickness is no longer small compared to r0. This result is not practically important, since the fluxes for 6 close to tt are relatively small and hardly count in the total transport rate. It may also be noted that, as a consequence of the approximations in the derivation, the limit v —> 0 of equation (45) does not approach the purely diffusional value given by ro. 

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