APPENDIX D: Direct and Indirect Band-Gap and Optical Transitions
In chapter 2 we have seen that, as far as the band-gap is concerned, there are two kinds of semiconductors, those in which the conduction band minimum Ec and the valence band maximum Ev are at the same к point (called direct band-gap semiconductors), and those in which Ec and Ev are at different к points (called indirect band-gap semiconductors). A typical example of the first kind is GaAs, in which both Ey and Ec are at к = 0, and of the latter is Si in which Ey lies at A: = 0 but Ec is six-fold degenerate and lies somewhere along such axes as (100). The two types of E(k) curves are shown schematically in figure Dl.
This difference in band-gaps is of a paramount importance for their optical properties. In short, indirect band-gap semiconductors cannot exhibit optical emission whereas direct semiconductors do and are the basic materials for the manufacturing of LASERs and LEDs. Both the interaction of an incident photon with an electron and the emission of a photon by an excited electron can be considered as a scattering process in which the energy and the momentum are conserved.
For a relaxation process in which an excited electron at Ec relaxes to Ev by emission of a photon of energy ftco we have
where p(Ec) is the momentum of the excited electron at Ec, p(Ev) is the momentum of the electron at Ey after relaxation and p(photon) the momentum of the emitted photon. Equation D2 may be written
But photons carry momentum: according to the De Broglie hypothesis, electrons and photons are symmetrical, both have momentum p and wavelength A, related to each other by p = 2пЫХ
FIGURE D1 Schematic E (k) band diagrams showing (a) a direct and (b) an indirect band-gap semiconductor. Allowed and forbidden transitions are also shown.
(in 1 dimension). It is worth pointing out that even classically the electromagnetic field has momentum. Therefore
where X is the wavelength of the photon.
However, the wavevectors of electrons are of the order nla where a is the interatomic distance, a few Angstroms, whereas the wavelength of visible light is of the order of 10 2nm ~ 103A. Hence the change |Д/с| of an electron must be negligible. Obviously in our argument the implicit assumption that the electron has not taken up momentum from the lattice has been made. This restriction (that |Д/с| = 0) leads to what is called a vertical transition and has important consequences for the emission process in a semiconductor. If an electron finds itself in the conduction band minimum Ec of a direct band-gap semiconductor it can relax by a vertical transition to the valence band maximum Ey because empty states are available there (assuming the semiconductor is doped by acceptors), see
FIGURE D2 Energy and momentum conservation during an electron relaxation from the conduction to the valence band.
the arrows in figure Dla. But if the excited electron is in an indirect semiconductor it cannot relax by a vertical transition because it will find itself in occupied states, see the arrow in figure Dlb. Note that this transition is not allowed because of the combined effect of two laws, conservation of momentum and the Pauli’s principle.
Note that the reverse transition (i.e. the absorption of a photon by the excitation of an electron) in an indirect semiconductor is not forbidden: an electron can go from an occupied state in the valence band to an empty state in the conduction band by a vertical transition. However, we have emphasized the emission process because this process forms the basis for the operation of LEDs and LASERs. In these devices the electrons find themselves in excited states because of the application of an external voltage.