# The Theory of Scanning Quantum Dot Microscopy

## TL;DR

This paper introduces a comprehensive theoretical framework for scanning quantum dot microscopy (SQDM), enabling quantitative analysis of surface potentials at the nanoscale across various sample types.

## Contribution

It develops a general electrostatic boundary value problem-based theory for SQDM, applicable to diverse sample properties and facilitating precise interpretation of imaging data.

## Key findings

- Established a formalism for quantitative surface potential imaging.
- Unified theory applicable to conductive and insulating nanostructures.
- Enhanced understanding of electrostatic interactions in SQDM.

## Abstract

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.