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Confined electron systems, such as 2D electron gases (2DEGs), 2D materials,or topological insulators show great technological promise but theirsusceptibility to defects often results in nanoscale inhomogeneities withunclear origins. Scattering-type scanning near-field optical microscopy(s-SNOM) is useful to investigate buried confined electron systemsnon-destructively with nanoscale resolution, however, a clear separation ofcarrier concentration and mobility was often impossible in s-SNOM. Here, wepredict a previously inaccessible characteristic "fingerprint" response of theprototypical LaAlO$_3$/SrTiO$_3$ 2DEG, and verify this using a state-of-the-arttunable narrow-band laser in mid-infrared cryo-s-SNOM at 8 K. Our modellingallows us to separate the influence of carrier concentration and mobility onfingerprint spectra and to characterize 2DEG inhomogeneities on the nanoscale.This spatially resolved information about the local electronic properties canbe used to identify the origin of inhomogeneities in confined electron systems,making the s-SNOM fingerprint response a valuable tool for nanoelectronics andquantum technology.

Mid infrared near-field fingerprint spectroscopy of the 2D electron gas in LaAlO$_3$/SrTiO$_3$ at low temperatures