Infrared study of the quasi-two-dimensional electron system at the interface between SrTiO$_{3}$ and crystalline or amorphous LaAlO$_3$

We have used grazing-angle infrared spectroscopy to detect the Berreman effect (BE) in the quasi-two-dimensional electron system (q-2DES) which forms spontaneously at the interface between SrTiO$_{3}$ (STO) and a thin film of LaAlO$_3$ (LAO). From the BE, which allows one to study longitudinal optical excitations in ultrathin films like the q-2DES, we have extracted at different temperatures its thickness, the charge density and mobility of the carriers under crystalline LAO (sample A), and the charge density under amorphous LAO (sample B). This quantity turns out to be higher than in sample A, but a comparison with Hall measurements shows that under amorphous LAO the charges are partly localized at low $T$ with a low activation energy (about 190 K in $k_B$ units), and are thermally activated according to a model for large polarons. The thickness of the q-2DES extracted from our spectra turns out to be 4 $pm 1$ nm for crystalline LAO, 7 $pm 2$ nm for amorphous LAO.