  Electrostatic forces are among the most common interactions in nature and
omnipresent at the nanoscale. Scanning probe methods represent a formidable
approach to study these interactions locally. The lateral resolution of such
images is, however, often limited as they are based on measuring the force
(gradient) due to the entire tip interacting with the entire surface. Recently,
we developed scanning quantum dot microscopy (SQDM), a new technique for the
imaging and quantification of surface potentials which is based on the gating
of a nanometer-size tip-attached quantum dot by the local surface potential and
the detection of charge state changes via non-contact atomic force microscopy.
Here, we present a rigorous formalism in the framework of which SQDM can be
understood and interpreted quantitatively. In particular, we present a general
theory of SQDM based on the classical boundary value problem of electrostatics,
which is applicable to the full range of sample properties (conductive vs
insulating, nanostructured vs homogeneously covered). We elaborate the general
theory into a formalism suited for the quantitative analysis of images of
nanostructured but predominantly flat and conductive samples.
