**1. Create a model of charge distribution in material:**

For example, a charged grain boundary normal to the surface can be modeled as a plane with a given interface charge density compensated by two uniformly charged depletion regions (abruptjunction approximation). A double Schottky barrier model

allows an analytical expression for the stray fields in air above surface interface junction. A similar example is a ferroelectric surface, which is traditionally represented as a constant charge density (unscreened surface) or a constant dipole moment density (completely screened surface) within domains with rapid variations of surface properties at the domain boundary.

** 2. Describe stray fields in air:**

Once the interface model is established, it can be used to calculate the field distribution above the surface. For some cases this can be done analytically; however, in general it requires numerical integration or Finite Element Analysis (FEA). For surfaces with lateral inhomogeneities, first order perturbation theory can often be used. In this particular case, the potential above the grain boundary-surface junction iswhere dsc is the depletion width and z is the distance from the surface. j0 is the potential at the grain boundary-surface junction, which is equal to grain boundary potential in the bulk for an unscreened surface if a high dielectric constant material.

Calculate the surface-tip interaction

**3. Calculate the surface-tip interaction:**

Provided the potential and/or electric field in air above the surface is known (step 2), its effect on the static and dynamic properties of a cantilever can be established. Different non-contact SPM techniques are sensitive to different interactions, e.g. Electrostatic

Force Microscopy (EFM) detects the force gradient acting on the tip, while Scanning Surface Potential Microscopy (SSPM) detects the first harmonic of the force. The difference in imaging mechanisms provides independent sets of data. Unfortunately, contrast in SSPM can be sensitive to imaging parameters, e.g. driving voltage in SSPM, feedback gains, etc. Hence the influence of all parameters on SSPM image must be taken into account to be quantitative. For an electroactive grain boundary, the total force gradient acting on the tip is

where the first term is the capacitive force between the tip and dielectric surface

and the second term is coulombic term due to grain boundary charges

From these formulae the total force and force gradient acting on the tip can be calculated and used to describe experimental data.

**4. Quantification of SSPM images in terms of the model and extraction of model parameters**

If the stray field distribution in air and the probe response to the fields is known (step 2 and 3), SSPM images can be interpreted in terms of the local field. Acquisition of several images at different tip biases (EFM) allows different contributions into image contrast, i.e.

capacitive (parabolic in bias) and Coulombic forces (linear in bias), to be distinguished. Performing EFM and SSPM imaging at different tip-surface separations allows reconstruction of the distance dependence of the forces. Clearly, the decay length of the stray fields is determined by the specifics of the charge distribution in the material. For example, for charged grain boundaries the decay length is comparable to the depletion width. Mathematical analysis of distance dependence yields model parameters and the local material properties.

**5. Environmental conditions:**

Presence of mobile surface charges can effectively screen the bulk charges. The most spectacular example of such behavior is on ferroelectric surfaces, but screening can occurs in other systems as well and we believe it to be a rather universal phenomenon.