Set of supplementary notes.

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.