Semiconductors charge distribution inside

The Diffuse-Charge Region inside an Intrinsic Semiconductor The Garrett-Brattain Space Charge. After this elementary account of the constitution of an intrinsic semiconductor, one can consider the basic question posed in Section 6.10.1.1. Given a layer of charge on the OUP of the electrolyte, how do the electrons and holes inside an intrinsic semiconductor distribute themselves as a function of distance from the interface ... 

Garrett and Brattain were the first to attack this problem, and they relied upon the similarity to the distribution of ions in solution when they considered the distribution of electrons and holes inside semiconductors. Thus, deep inside an intrinsic semiconductor the excess charge density must be zero because of the equality of electrons and holes. 

Since the metal can be treated as a nearly perfect conductor, C is high compared with C, and cannot influence the value of the measured doublelayer capacitance. The role of the metal in the double layer structure was discussed by Rice, who suggested that the distribution of electrons inside the metal decides the properties of the double-layer. This concept was later used to describe double-layer properties at the semiconductor/electrolyte interface. As shown later, the electron density on the metal side of the interface can be changed under the influence of charged solution species (dipoles, ions). ... 

For a semiconductor like Ge, the pattern of electronic interaction between the surface and an adsorbate is more complex than that for a metal. Semiconductors possess a forbidden gap between the filled band (valence band) and the conduction band. Fig. 6a shows the energy levels for a semiconductor where Er represents the energy of the top of the valence band, Ec the bottom of the conduction band, and Ey is the Fermi energy level. The clean Ge surface is characterized by the presence of unfilled orbitals which trap electrons from the bulk, and the free bonds give rise to a space-charge layer S and hence a substantial dipole moment. Furthermore, an appreciable field is produced inside the semiconductor, as distinct from a metal, and positive charges may be distributed over several hundred A. 

Adsorption. In the simple theory of the space charge inside a semiconductor, it was assumed that all the electrons and holes are free to move up to the surface. Being susceptible to thermal motion, their concentrations from a- = 0 to x — °° were said to be given by the interplay of electrical and thermal forces only, as expressed by the Boltzmann distribution law and Poisson s equation. 

Consider the interfacial region shown in detail in Fig. 4. The total interfacial drop, ,-, is composed of three contributions 8C, the space-charge potential dropped inside the semiconductor, potential across the (uncharged) Helmholtz layer, and el, the potential dropped in the electrolyte. To solve for the potential distribution in the interfacial region, we make use of... 

Metal electrodes have been so extensively studied in the past that one may profitably ask the question How does a semiconductor electrode differ from a metal electrode Although there are many differences, the two essential distinctions are (1) the low and readily variable electron density and (2) the existence of an energy gap, i. e. of a region of unallowed electronic energies. The many other differences can be shown to arise from these two. For example, because of the low electron density, one has sizable penetration of the interfacial electric fields into the bulk of the electrode. With metals significant fields penetrate no more than an angstrom or so inside the electrode with typical semiconductors the fields may be sizable at depths in excess of 10,000 A. In consequence the distributions of charge and potential at a semiconductor electrode are almost of a different kind from those at a metal electrode. 

It is not surprising that the Poisson-Boltzmann approach has been used frequently in computing interactions between charged entities. Mention may be made of the Gouy theory (Fig. 3.24) of the interaction between a charged electrode and the ions in a solution (see Chapter 6). Other examples are the distribution (Fig. 3.25) of electrons or holes inside a semiconductor in the vicinity of the semiconductor-electrolyte interface (see Chapter 6) and the distribution (Fig. 3.26) of charges near a polyelectrolyte molecule or a colloidal particle (see Chapter 6). 

Suppose that a semiconductor of thickness L is contacted with an electrode that, hy virtue of a low-energy barrier at the interface, is able to supply an unlimited number of one type of carrier. The current is then limited by its own space charge which, in die extreme case, reduces the electric field at the injecting contact to zero. This is realized when the number of carriers per unit area inside the sample approaches the capacitor charge, i.e. sso/e. It is this number of carriers dial can be transported per transit time ttr=d/fji. Hence, the maximum current is iscL = s,E,QfiF ld. A more rigorous treatment has to take into account the non-uni-form distribution of space charge and, concomitantly, electric field [34]. Starting with Poisson s equation and the continuity equation. 

Furthermore from the measured bond lengths we can deduce that inside each tetrad the two central TCNQ molecules (A and A on the fig. 5) bear less charge than the two others. This distribution gives an explanation for the paramagnetic and semiconductor behaviour. 

Eventually the system relaxes to the distribution and (f>2 value obtained by the GCS approach (Section 13.3). A similar simulation approach was taken to calculate the charge and potential distributions in the space-charge region that forms inside the electrode at a semiconductor electrode/electrolyte interface (20) (See Section 18.2). Migrational effects in voltammetry can be taken into account in a similar way by using the full Nernst-Planck equation to treat mass transfer (21). 

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