Near-field spectroscopy of a gated electron gas

We study the spatial distribution of the photoluminescence of a gated two-dimensional electron gas with sub-wavelength resolution. This is done by scanning a tapered optical fibre tip with an aperture of 250 nm in the near field region of the sample surface, and collecting the photoluminescence. The spectral line of the negatively charged exciton, formed by binding of a photo-excited electron-hole pair to an electron, serves as an indicator for the local presence of charge. The local luminescence intensity of this line is directly proportional to the number of electrons under the tip. We observe large spatial fluctuations in this intensity in the gate voltage range, where the electron conductivity exhibits a sharp drop. The amplitude of these fluctuations increases and the Fourier spectrum extends to lower spatial frequencies as the gate voltage becomes more negative. We show that the fluctuations are due to the statistical distribution of localised electrons in the random potential of the remote ionised donors. We use these fluctuations to image the electron and donor distribution in the plane.