  Probing the local transport properties of two-dimensional electron systems
(2DES) confined at buried interfaces requires a non-invasive technique with a
high spatial resolution operating in a broad temperature range. In this paper,
we investigate the scattering-type scanning near field optical microscopy as a
tool for studying the conducting LaAlO3/SrTiO3 interface from room temperature
down to 6 K. We show that the near-field optical signal, in particular its
phase component, is highly sensitive to the transport properties of the
electron system present at the interface. Our modelling reveals that such
sensitivity originates from the interaction of the AFM tip with coupled
plasmon-phonon modes with a small penetration depth. The model allows us to
quantitatively correlate changes in the optical signal with the variation of
the 2DES transport properties induced by cooling and by electrostatic gating.
To probe the spatial resolution of the technique, we image conducting
nano-channels written in insulating heterostructures with a voltage-biased tip
of an atomic force microscope.
