Polaron effect in GaAs-Ga 1¿x Al x As quantum wells: A fractional-dimensional space approach

The binding energy and the effective mass of a polaron confined in a $\mathrm{GaAs}\ensuremath{-}{\mathrm{Ga}}_{1\ensuremath{-}x}{\mathrm{Al}}_{x}\mathrm{As}$ quantum well are calculated within the framework of the fractional-dimensional space approach. In this scheme, the real confined ``polaron plus quantum well'' system is mapped onto an effective fractional-dimensional bulk in which the polaron behaves unconfined, and the fractional dimension is essentially related to the degree of confinement of the actual system. Analytical expressions allowing a very simple estimation of the corresponding polaron corrections are found. The fractional-dimensional theoretical results are shown to be in good agreement with previous more detailed calculations.