Generalized superlattice K·p theory and intersubband optical transitions

Abstract

The eight-band superlattice crystal K·p formalism is extended to include the higher-lying antibonding $p$ states perturbatively. The initial 14×14 Hamiltonian matrix is block diagonalized (or folded down) so that corrections due to these states appear in the modified 8×8 matrix. Their principal effect is to introduce a finite heavy-hole mass. The use of the eight-band formalism for band-structure and interband optical calculations is validated because other corrections are very small. The approximate wave functions associated with the original 14×14 Hamiltonian are used to calculate intersubband optical matrix elements. These include new terms, present in crystals lacking inversion symmetry, which increase the magnitude of conduction intersubband absorption by about three orders of magnitude for in-plane polarized (TE) photons, while leaving absorption due to growth-axis polarized (TM) photons relatively unaffected. The TE absorption in narrow $\mathrm{GaAs}/{\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ quantum wells (QW's) is shown to be observable. However, use of the $f$-sum rule and the cyclotron-resonance-determined conduction-band effective-mass anisotropy indicate an upper limit of about 20% relative to TM absorption. Comparison with the measured TM absolute absorption coefficient for a system used for quantum-well infrared photoconductors (QWIP's) yields excellent agreement. The conductivity sum rule involving these intersubband transitions is used to confirm the observed broadening associated with very small QW width fluctuations for this system.