Signatures of non-local conductivity in near-field microscopy

We propose and theoretically substantiate a new method to study the non-local conductivity of two-dimensional electron systems (2DES) using the tools of near-field microscopy. We show that the height dependence of induced dipole moment of illuminated near-field probe is substantially different for various transport regimes of charge carriers in 2DES. For hydrodynamic transport regime, the induced dipole moment scales as $z_0^{-2}$, where $z_0$ is the elevation of probe above the 2DES. Both for Drude and classical ballistic regimes of conduction, the dipole moment scales as $z_0^{-3}$. In the former case, the dipole moment is carrier density-independent, while in the latter it largely depends on carrier density. More generally, we find that the induced dipole moment of the probe is proportional to the Laplace transform of wave-vector dependent conductivity and inverse dielectric function of 2DES over the wave vectors $q$. Our results should provide a simple tool for studies of non-local conductivity in solids that was challenging to address with other techniques.