Two-Photon Absorption in an Indirect-Gap Semiconductor Quantum Well System. II. Excitonic Transitions

A theory of phonon-assisted two-photon exciton transitions in indirect-band-gap semiconductor layered quantum wells (QW) and quantum well wires (QWW) is developed. The expressions for the two-photon absorption coefficients in one-dimensional (1D), α,2D, α, and (bulk), α, respectively, are calculated. The spectral dependence of these expressions, the vicinity of the band edge, are found to obey the law (2ℏω ± ℏΩ – EG + E)β, where ℏω (ℏΩ) is the photon (phonon) energy, EG the effective indirect gap, and Ek the exciton binding energy. The values of β vary from – ½ up to ½ depending on the dimension of the system and the type of the coupling matrix element involved in each transition process. Before the edge, the final exciton states are of s- or p-symmetry according to the photon polarizations with respect to the confinement directions in the QWs and to the selection rules allowed by the momentum matrix elements. A numerical estimation for the case Si0.5Ge0.5 shows that α for both photon polarizations (parallel and perpendicular to the confinement direction of the QW's) is enhanced over the values of α and α (allowed and forbidden transitions). Furthermore α is also enhanced over the α values of bulk materials. This behaviour of the αex's is interpreted as due to (i) the additional confinement of the carriers which occur going from 3D 2D 1D systems, (ii) the photon polarization configurations with respect to the confinement directions, and (iii) the coupling matrix elements.