from the conduction to the valence band and a photon is emitted. Direct-gap semiconductors(e.g, GaAs)are widely used in optoelectronics. In indirect-band materials [e.g, Si, see Fig 22. 3(a)], a band-to-band transition requires a change of momen- tum that cannot be accomplished by absorption or emission of a photon. Indirect band-to-band transitions require the emission or absorption of a phonon and are much less probable than direct transitions. For ho< E lie, for >A2=1.24 um/E (ev)-cutoff wavelength] band-to-band transitions do not occur, but light can be absorbed by a variety of the so-called subgap processes. These processes include the absorption by free carriers, formation of excitons(bound electron-hole pairs whose formation requires less energy than the creation of a free electron and a free hole), transitions involving localized states(e.g, from an acceptor state to the conduction band), and phonon absorption. Both band-to-band and subgap processes may be responsible for the increase of the free charge carriers concentration. The resulting reduction of the resistivity of illuminated semiconductors is called photoconductivity and is used in photodetectors. In a strong magnetic field(ot > 1)the absorption of microwave radiation is peaked at o=0, At this frequency the photon energy is equal to the distance between two Landau levels, ie,, ha= Es+I- Es with reference to Eq.(22. 13). This effect, known as cyclotron reson used to measure the effective masses of charge carriers in semiconductors [ in a simplest case of isotropic E(k)dependence, m*=q B/o In indirect-gap materials like silicon, the generation and annihilation(or recombination) of electron-hole pairs is often a two-step process. First, an electron(or a hole)is trapped in a localized state( called a recombi- nation center)with the energy near the center of the energy gap. In a second step, the electron(or hole)is transferred to the valence(conduction) band. The net rate of recombination per unit volume per unit time given by the Shockley-Read-Hall theory as R (22.16) t(P+Pi)+t, (n+n) where t,, te, Pi, and n, are parameters depending on the concentration and the physical nature of recombination centers and temperature. Note that the sign of r indicates the tendency of a semiconductor toward equilibrium (where np ni, and R=O). For example, in the depleted region np f and R< 0, so that charge carriers ar generated shockley-Read-Hall recombination is the dominating recombination mechanism in moderately doped silicon. Other recombination mechanisms(e.g, Auger) become important in heavily doped semiconductors Wolfe et al., 1989; Shur, 1990; Ferry, 1991] The recombination processes are fundamental for semiconductor device theory, where they are usually an diy )n-r (22.17) q Nanostructure engineeri Epitaxial growth techniques, especially molecular beam epitaxy and metal-organic chemical vapor deposition, allow monolayer control in the chemical composition process. Both single thin layers and superlattices can be obtained by such methods. The electronic properties of these structures are of interest for potential device applications. In a single quantum well, electrons are bound in the confining well potential. For example, in a rectangular quantum well of width b and infinite walls, the allowed energy levels are E、k)=π2s2h2/(2mxb2)+h2k2/(2mx),s=1,2,3, (22.18) where k is the electron wave vector parallel to the plane of the semiconductor layer. The charge carriers quantum wells exhibit confined particle behavior. Since E, o b-, well structures can be grown with distance e 2000 by CRC Press LLC© 2000 by CRC Press LLC from the conduction to the valence band and a photon is emitted. Direct-gap semiconductors (e.g., GaAs) are widely used in optoelectronics. In indirect-band materials [e.g., Si, see Fig. 22.3(a)], a band-to-band transition requires a change of momen￾tum that cannot be accomplished by absorption or emission of a photon. Indirect band-to-band transitions require the emission or absorption of a phonon and are much less probable than direct transitions. For \w < Eg [i.e., for l > lc = 1.24 mm/Eg (eV) – cutoff wavelength] band-to-band transitions do not occur, but light can be absorbed by a variety of the so-called subgap processes. These processes include the absorption by free carriers, formation of excitons (bound electron–hole pairs whose formation requires less energy than the creation of a free electron and a free hole), transitions involving localized states (e.g., from an acceptor state to the conduction band), and phonon absorption. Both band-to-band and subgap processes may be responsible for the increase of the free charge carriers concentration. The resulting reduction of the resistivity of illuminated semiconductors is called photoconductivity and is used in photodetectors. In a strong magnetic field (wct >> 1) the absorption of microwave radiation is peaked at w = wc. At this frequency the photon energy is equal to the distance between two Landau levels, i.e., \w = ES+1 – ES with reference to Eq. (22.13). This effect, known as cyclotron resonance, is used to measure the effective masses of charge carriers in semiconductors [in a simplest case of isotropic E(k) dependence, m* n = qB/wc]. In indirect-gap materials like silicon, the generation and annihilation (or recombination) of electron–hole pairs is often a two-step process. First, an electron (or a hole) is trapped in a localized state (called a recombi￾nation center) with the energy near the center of the energy gap. In a second step, the electron (or hole) is transferred to the valence (conduction) band. The net rate of recombination per unit volume per unit time is given by the Shockley–Read–Hall theory as (22.16) where tn, tp, p1, and n1 are parameters depending on the concentration and the physical nature of recombination centers and temperature. Note that the sign of R indicates the tendency of a semiconductor toward equilibrium (where np = n2 i , and R = 0). For example, in the depleted region np < n2 i and R < 0, so that charge carriers are generated. Shockley–Read–Hall recombination is the dominating recombination mechanism in moderately doped silicon. Other recombination mechanisms (e.g., Auger) become important in heavily doped semiconductors [Wolfe et al., 1989; Shur, 1990; Ferry, 1991]. The recombination processes are fundamental for semiconductor device theory, where they are usually modeled using the continuity equation (22.17) Nanostructure Engineering Epitaxial growth techniques, especially molecular beam epitaxy and metal-organic chemical vapor deposition, allow monolayer control in the chemical composition process. Both single thin layers and superlattices can be obtained by such methods. The electronic properties of these structures are of interest for potential device applications. In a single quantum well, electrons are bound in the confining well potential. For example, in a rectangular quantum well of width b and infinite walls, the allowed energy levels are Es (k) = p2s2\2/(2m* n b 2) + \2k2/(2m* n), s = 1, 2, 3, . . . (22.18) where k is the electron wave vector parallel to the plane of the semiconductor layer. The charge carriers in quantum wells exhibit confined particle behavior. Since Es } b –2, well structures can be grown with distance R np n p p n n i n p = - + + + 2 1 1 t ( ) t ( ) ¶ ¶ n t div q R n = - j