  We demonstrate a dynamic scanning capacitance microscope (DSCM) that operates
at large bandwidths, cryogenic temperatures and high magnetic fields. The setup
is based on a non-contact atomic force microscope (AFM) with a quartz tuning
fork sensor with non-optical excitation and read-out for topography, force and
dissipation measurements. The metallic AFM tip forms part of an rf resonator
with a transmission characteristics modulated by the sample properties and the
tip-sample capacitance. The tip motion gives rise to a modulation of the
capacitance at the frequency of the AFM sensor and its harmonics, which can be
recorded simultaneously with the AFM data. We use an intuitive model to
describe and analyze the resonator transmission and show that for most
experimental conditions it is proportional to the complex tip-sample
conductance, which depends on both the tip-sample capacitance and the sample
resistivity. We demonstrate the performance of the DSCM on metal disks buried
under a polymer layer and we discuss images recorded on a two-dimensional
electron gas in the quantum Hall effect regime, i.e. at cryogenic temperatures
and high magnetic fields, where we directly image the formation of compressible
stripes at the physical edge of the sample.
