Nonparabolicity and a sum rule associated with bound-to-bound and bound-to-continuum intersubband transitions in quantum wells

A sum rule for electronic intersubband transitions has been derived following Kane’s model, beyond the quadratic dispersion relations. The sum rule takes into account the effects of nonparabolicity and the different effective masses in the well and barrier materials; it depends on the property of the ground state of the system and, as such, on the shape of the potential. The boundaries of the validity of matrix element computations are also discussed in the case where only the conduction band is included. Experimental results are presented for bound-to-bound and bound-to-continuum intersubband transitions in various types of ${\mathrm{Al}}_{\mathit{x}}$${\mathrm{In}}_{1\mathrm{-}\mathit{x}}$As/${\mathrm{Ga}}_{\mathit{y}}$${\mathrm{In}}_{1\mathrm{-}\mathit{y}}$As quantum well systems (single wells, coupled wells and quantum wells with Bragg confinement); the agreement with theory is excellent. In the last section of the paper, the effect of the electric field on the sum rule is investigated.