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120 CHAPTER 8. SEMICONDUCTOR DEVICES in general line up in an offset fashion. We consider here only the case (like (Al, Ga)As when the band edges of one semiconductor lie entirely within the band gap of the other, though staggered overlaps do occur. When the materials are placed in contact, their Fermi energies must equalise, which is accomplished by charge transfer across the boundary. This lowers the conduction band edge on one side of the interface, and if doped sufficiently the band edge falls below the chemical potential, so that an equilibrium electron gas forms at the interface. 8.5.2 Inversion layers Fig. 8.20 shows an outline of a scheme called modulation doping, where the donor levels are placed on the side of the interface away from the electron layers (and often at some distance from the interface). This has the advantage of creating an electron gas in a region where the crystal structure is nearly perfect, and mobilities greater than 10 3 m 2 V −1 s −1 have been achieved at low temperature. By addition of metal gates to the surface of the strucutures, electrical potential gradients can be applied to continuously vary the electron density in the layer, to pattern one dimensional wires, and to construct other interesting spatial structures. 8.5.3 Quantum wells One of the most widespread applications of semiconductor multilayers is to make a quantum well — a thin region of a narrow gap material sandwiched inside a wide-gap one. Because the wells can be made very narrow, quantisation of the levels is important. In general, the eigenstates will be of the form Φ(r, z) = φ n (z)e ik·r where r and k are here two-dimensional vectors, describing position and momentum in the plane. The situation for holes is more complex, because the degeneracy of the light and heavy hole states in bulk is broken by the 2D geometry. The details are important in practice, but not exciting. 8.5.4 Quantum well laser The operation of a laser requires an efficient mechanism for luminescent electron-hole recombination, which rules out indirect gap semiconductors in practice. Lasing operation requires high densities of electrons and holes so that the probability of stimulated emission overcomes that of absorption. This latter condition requires inversion, meaning that the average electron (hole) occupancy in the luminescing states exceeds 1/2. A double heterojunction laser is designed to achieve high densities, by using a quantum well — designed to trap both electrons and holes — with the source of carriers being a p-doped region on one side of the well, and an n-doped region on the other (see Fig. 8.21). This is indeed a diode (because holes can flow in from the p-side and electrons from the n, but not vice versa), but it is not operated in the same regime as a conventional diode. Instead, a rapid rate of recombination in the lasing region maintains different chemical potentials for electron and hole systems.
8.5. COMPOUND SEMICONDUCTOR HETEROSTRUCTURES 121 Figure 8.20: Formation of a 2D electron system by modulation doping. (a) Shows two semiconductors not in contact, with different chemical potentials, determined by the doping level. (b) is the band scheme that results when they are placed in contact. If the doping level is high enough - as shown here - the band edge on the left may fall below the chemical potential, and a layer of electrons is formed at the interface. (c) Shows the modulation doping scheme in more detail. Donors are placed to the right of the interface, so that the electron layer is pristine and free of impurities. The well width may be narrow enough that electron levels are quantised in a direction perpendicular to the barrier, forming sub-bands. Figure 8.21: Operation of a double heterojunction laser. Notice the quasi-equilibrium condition, with separate electron and hole chemical potentials.
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34 CHAPTER 3. FROM ATOMS TO SOLIDS
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