Electron-Hole Product

we need to find the number of electrons and holes present as a function of T. At thermal equilibrium, the number of electrons occupying the conduction band will be given by 

The density of states g Ec) is the same as our previous result (Equation 15.28) with the effective electron mass m substituted for m and E — Eq substituted for , or 

The electron-hole product is an important quantity that is given by 

This is sometimes referred to as the law of mass actior. Note that the number of charge carriers depends only on the effective masses, the temperature, and the energy gap. Also note that the Fermi level does not appear in this product, which means that no assumptions have been made concerning the origin of the electrons and holes. Therefore Equation 20.13 applies to electrons that have been extrinsically generated (from donor or collector states) as well as to intrinsic electrons and holes. As more electrons are added, some will go to annihilating holes until the equilibrium (Equation 20.13) is reached. 

If the material is intrinsic (no electrical active impurities), at equilibrium there must be a hole for every electron promoted to the conduction band. Therefore, ni = pi and 

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The overall efficiency of LED emission depends on three factors which vary between the different types of LEDs. These are the efficiency of electron-hole production, the radiative efficiency of recombination, and the efficiency of extraction of the optical signal from the junction. 

Electronic hole production and lone pair production have opposite effects on the DOS distribution between —3.0 eV and Ep. The framer weakens the DOS intensity, while the latter enhances it. What one can detect is the resultant of these two effects. The DOS change detected in this region may be insignificant if these two opposite processes are comparable in quantity. 

The electron-hole pairs produced by the absorption of photons can be detected by applying a potential across the semiconducting material and measuring the current. The resulting photocurrent is directly proportional to the rate of electron-hole production, which is proportional to the photon flux. Thus a semiconductor crystal provides a very simple device for measuring the intensity of light. 

Pure materials, or at least materials with no impurity states in the bandgap region, are called intrinsic semiconductors. The creation of an electron leaves a hole therefore, the number of holes must equal the number of electrons in an intrinsic material. The electron-hole product is directly proportional to the Boltzmann factor, exp —(Eg/fcT). The Fermi level, the energy for which the probability of creating a conduction electron is the same as creating a hole is shovm to be somewhere in the bandgap. 

We saw in Chapter 20 that the electron-hole product in an intrinsic semiconductor is given by np (2.5 x 10 /m ) exp(—Eg/fcT) so the number of charged carriers will be (5 X 10 /m ) exp( Eg/2fcT). Even with a bandgap as low as 3 eV, the number of carriers present at ambient temperature would be less than 1 electron or hole/m. However, if electrically active impurities are present that could act as donors or acceptors, the electron or hole population could increase dramatically. For example, 1 ppm of a donor impurity, if fully ionized, would produce a carrier concentration of 10 electrons/m. ... 

Another common loss process results from electron—hole recombination. In this process, the photoexcited electron in the LUMO falls back into the HOMO rather than transferring into the conduction band. This inefficiency can be mitigated by using supersensitizing molecules which donate an electron to the HOMO of the excited sensitizing dye, thereby precluding electron—hole recombination. In optimally sensitized commercial products, dyes... 

According to the electron-transfer mechanism of spectral sensitization (92,93), the transfer of an electron from the excited sensitizer molecule to the silver haHde and the injection of photoelectrons into the conduction band ate the primary processes. Thus, the lowest vacant level of the sensitizer dye is situated higher than the bottom of the conduction band. The regeneration of the sensitizer is possible by reactions of the positive hole to form radical dications (94). If the highest filled level of the dye is situated below the top of the valence band, desensitization occurs because of hole production. 

If for example Ti02, is used to capture sunlight in a photo-catalytic reaction then only about 10% of the available spectrum will be of use, since it requires 3.2 eV to create an electron-hole pair in Ti02. Both the photovoltaic and the photochemical methods are of potential interest, but at present they are too expensive. Also, the production of semiconductors used in photovoltaic cells consumes much energy. Nevertheless, the prospect remains attractive. If cells could be made with an efficiency of say 10 % then only 0.1 % of the earths surface would be required to supply our present energy consumption ... 

The photoreactivity of the involved catalyst depends on many experimental factors such as the intensity of the absorbed light, electron-hole pair formation and recombination rates, charge transfer rate to chemical species, diffusion rate, adsorption and desorption rates of reagents and products, pH of the solution, photocatalyst and reactant concentrations, and partial pressure of oxygen [19,302,307], Most of these factors are strongly affected by the nature and structure of the catalyst, which is dependent on the preparation method. The presence of the impurities may also affect the photoreactivity. The presence of chloride was found to reduce the rate of oxidation by scavenging of oxidizing radicals [151,308] ... 

For a single band gap system, two basic approaches for the conversion of hot carriers into electricity or chemical energy have been proposed to enhance the efficiency of photon conversion (a) extraction of the hot carries before they cool, with the production of an enhanced photovoltage [35] (b) production of two or more electron-hole pairs per photon absorbed, with photocurrent enhancement [36, 37]. 

Both electrons and holes are mobile charge carriers in semiconductors. The mobile charge carrier whose concentration is much greater than the other is called the majority carrier, and the minority carrier is in much smaller concentrations. In n-type semiconductors, the mcgority carriers are electrons in the conduction band and the minority carriers are holes in the valence band. The product of the concentrations of majority and minority carriers (electrons and holes) in a semiconductor of extrinsic type (containing impurities) equals the square of the concentration of electron-hole pairs, ni, in the same semiconductor of intrinsic type (containing no impurities) ... 

In this type of cell both electrodes are immersed in the same constant pH solution. An illustrative cell is [27,28] n-SrTiOs photoanode 9.5-10 M NaOH electrolyte Pt cathode. The underlying principle of this cell is production of an internal electric field at the semiconductor-electrolyte interface sufficient to efficiently separate the photogenerated electron-hole pairs. Subsequently holes and electrons are readily available for water oxidation and reduction, respectively, at the anode and cathode. The anode and cathode are commonly physically separated [31-34], but can be combined into a monolithic structure called a photochemical diode [35]. 

In addition to the dark oxidation of S(IV) on surfaces, there may be photochemically induced processes as well. For example, irradiation of aqueous suspensions of solid a-Fe203 (hematite) containing S(IV) with light of A > 295 nm resulted in the production of Fe(II) in solution (Faust and Hoffmann, 1986 Faust et al., 1989 Hoffmann et al., 1995). This reductive dissolution of the hematite has been attributed to the absorption of light by surface Fe(III)-S(IV) complexes, which leads to the generation of electron-hole pairs, followed by an electron transfer in which the adsorbed S(IV) is oxidized to the SO-p radical anion. This initiates the free radical chemistry described earlier. 

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