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

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