Width of the Depletion Layer

The electric field may be foimd from one of the Maxwell s equation, V-D = V-sE = p, 

Integrating again with the boundary condition that requires the potential to vanish at Xn, 

A similar argument can be made for the width of the depletion zone in the p-side. Note that the width of the depletion zone varies as the inverse square root of the doping level. Using the same doping level as above and taking the dielectric constant to be 12, the width of the depletion zone into the n-side is 

Thus we see that the depletion layers are extremely thin, on the order of tens of nanometers. Putting this width into Equation 21.12, we obtain an electric field of 5.54 x 10 V/m. 

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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). 

The speed of response of the photodiode depends on the diffusion of carriers, the capacitance of the depletion layer, and the thickness of the depletion layer. The forward bias itself increases the width of the depletion layer thus reducing the capacitance. Nevertheless, some design compromises are always required between quantum efficiency and speed of response. The quantum efficiency of a photodiode is determined largely by the absorption coefficient of the absorbing semiconductor layer. Ideally all absorption should occur in the depletion region. This can be achieved by increasing the thickness of the depletion layer, but then the response time increases accordingly. 

The space charge layer capacitance is inversely proportional to the width of the depletion layer w. As the width of the depletion layer approaches zero the capacitance approaches infinity, hence... 

Another possible effect of PdAu deposits on PdAu/SnOx sensors is through the formation of a Schottky barrier between PdAu and SnOx, as in the case of the Pd/CdS hydrogen sensor. If such a barrier is formed, then a depletion layer is created inside the semiconductor tin oxide. Since the Pd work function can be reduced by hydrogen absorption through dipole or hydride formation (14,15), the width of the depletion layer in tin oxide may be reduced. The reduction of the depletion layer width causes the sample resistance to decrease. Such a possibility was checked and was ruled out, because a good ohmic contact was obtained between Pd (-50 nm thick) and SnOx- It is also commonly known that gold forms an ohmic contact with tin oxide. 

In an ideal Schottky barrier, the width of the barrier is controlled by the width of the depletion layer (i.e., the concentration of dopants). The limiting factor in this case is the barrier height... 

Fig. 8.4. Profile of light intensity at the semiconductor electrolyte junction. W is the width of the depletion layer and L, is the hole diffusion length. The penetration depth of the light...

Fig. 5. Calculated interfacial potentials for (a) a moderately highly doped sample of n-GaP (Nt) = 10 cm 3, r,sc = 10) (b) a highly doped sample of n-Ti02 (Nu = 1019 cm-3, esc = 200) (c) an intrinsic sample of Ge (Ee 0.6 eV) (for all samples, dH = 3 A, cH = 6, el = 80 and C = 1 M) and (d) behaviour of the Fermi level and band edges for case (a) shown diagram-matically (IV defines the width of the depletion layer).

Numerical analysis shows that, for these small values of rj or, equivalently, large values of x, a good approximation for A is k exp (2 Y — 1)1/2 where k lies in the range 0.60-0.62. Numerical analysis also shows that the approximation (36) for the width of the depletion layer corresponds to the point at which the potential drop (/> — (pt has decreased to a value of 0.01 V, accurately enough zero for most purposes. 

The structure of a p-4-n device is shown in Fig. 10.1. The depletion layer width of low defect density undoped a-Si H at zero bias is about 1 pm, but is less than 100 A in heavily doped material (see Fig. 9.9). The p and n layers provide the built-in potential of the junction but contribute virtually nothing to the collection of carriers. Therefore the doped layers need be no more than the width of the depletion layer to establish the junction-any additional thickness unnecessarily reduces the charge collection by absorbing incident light. An efficient sensor usually requires that the undoped layer be as thick as possible to absorb the maximum flux of photons, but it cannot be thicker than... 

Because the spacing between pores is always less than the width of the depletion layer and PS has a very high resistivity, Beale et al. proposed that the material in the PS is depleted of carriers and the presence of a depletion layer is responsible for current localization at pore tips where the field is intensified. This intensification of field is attributed to the small radius of curvature at the pore tips. For lowly doped p-Si the charge transfer is by thermionic emission and the small radius of curvature reduces the height of the Schottky barrier and thus increases the current density at the pore tips. For heavily doped materials the current flow inside the semiconductor is by a tunneling process and depends on the width of the depletion layer. In this case the small radius of curvature results in a decrease of the width of the depletion layer and increases the current density at pore tips. The initiation was considered to be associated with the surface inhomogeneities, which provide the initial localized high current density at small surface depressions. 

The model of Beale et al. provided a deeper level of understanding of the current localization required for PS formation on different silicon substrates and pointed out the correlation between the relative dimension of pore size and the width of the depletion layer. Several concepts proposed in their model would be adopted and further developed in many of the later models such as those by FoU, Zhang, and Lehmann. " However, because the model considered only the physical aspects of the semiconductor and none of the chemical reactions, it provided little insight for the change of pore size and other morphological features with current and HF concentration. Also, Beale s model assumed that the Fermi level of the semiconductor is pinned on the surface on the midgap which does not agree with the later experimental data. 

Fig. 4. Energy diagram for a bulk monocrystalline n-type semiconductor electrode in the dark, in equilibrium with a redox system that has an equilibrium potential U. The Fermi-level (Ep) and the energy of the band edges are shown as a function of distance, x, perpendicular to the surface. The electrode is depleted of majority caniers at its surface, dsc is the width of the depletion layer and is the potential drop over the depletion layer.

It is useful to estimate the characteristic time required to achieve primary separation of the electron-hole pair in a bulk single crystalline semiconductor electrode with a flat interface with the electrolyte. If there is a depletion layer at the interface in which an electron-hole pair is generated (see Fig. 5), the electron will move to the interior and the hole to the surface. Provided that the mean free path length for electron-lattice scattering is much smaller than the width of the depletion layer, the drift velocities of the electron and hole are described by the relationships... 

M-S is an electrochemical impedance spectroscopy (EIS) technique [10-12] that can be difficult to perform and interpret if the system is not ideal. When the measurement is successful, it is able to determine both the fb and the free charge carrier density (donors or acceptors, A/Dopant) of the photoelectrode. Efb, along with the band gap (Eg) and the A dopant. can be used to determine the band structure of a photoelectrode and if it possesses the proper alignment with respect to the water splitting potentials (see chapter Introduction ). The A dopant also plays a role in the bulk and surface semiconductor properties such as the width of the depletion layer and rate of recombination. The conductivity type is also revealed by M-S analysis. The M-S plot will possess a negative slope for p-type materials and a positive slope for n-type materials (positive slope). In the case that the M-S measurement is not successful, then other techniques such as Hall Effect can still yield conductivity and A dopant for materials which can be deposited onto non-conductive substrates such as quartz. 

This entails that the width of the depletion layer h must be less than the width of the electrode d. 

Fig. 1 A scheme of the energetics at an n-type semiconductor electrode in contact with a redox system in an electrolyte solution, (a) The situation under conditions of electronic equilibrium. The electrochemical potential of the electrons is the same in both phases, i.e. the electron Fermi-level in the semiconductor Ep.n has the same value as the Fermi-level of the electrons in the redox system fRed/Ox- (t>) Case in which the energy bands in the semiconductor are flat this situation, corresponding to maximum photovoltage, is reached under strong illumination at open circuit. Wsc is the width of the depletion layer and e(j>sc is the band-bending.

Erne and coworkers measured the interfacial capacitance of macroporous GaP electrodes as a function of the electrode potential [138, 139). It was found that the capacitance is large for sufficiently small band-bending (interfacial layer in the porous solid) and decreases to the value of a nonporous interface at larger band-bending. Similar effects have been found with macroporous SiC and Si electrodes [18, 141). In fact, the interfadal capacitance is a measure for the surface area of the macroporous network, with the width of the depletion layer, VTsc. as a measuring stick. 

In addition to the value of the range resistor Ri, other parameters also affect the effective bandwidth of the system. As shown in Fig. 3.16b, the bandwidth improves when the width of the depletion layer (W) increases. This is because a larger value... 

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