FM-KPFM TheoryAn AC voltage with amplitude VAC and frequency fm (angular frequency ωm) superimposed on a DC voltage, VDC, is applied between the probe tip and the sample. The resulting electrostatic force is given by
where
where Δφ is the contact potential difference between the probe and sample.
Equation 1 may be separated into three terms:
The electric field gradient is given by:
The applied AC voltage modulates the force and force gradient at frequencies ωm and 2ωm.
Frequency Modulation
Hooke's law states:
Taking the derivative,
The force gradient and spring constant are thus seen to be equivalent.
The electrostatic force shifts the resonant frequency of a cantilever with effective mass m* as follows:
The applied AC voltage modulates Fel and ∂Fel/∂z according to Equation 4. Equation 6 then shows that the mechanical resonance of the cantilever is modulated with frequencies fm and 2fm with sidebands, shown schematically in Figure 8, appearing at f0±fm and f0±2fm.
Figure 8: Schematic of the frequency spectrum of the probe tip oscillation
Resolution
Figure 9: Charged sphere at separation z above an infinite plane
The force between a charged sphere at separation z above an infinite plane, shown in Figure 9, is:
Equation 10:
and its derivative is:
Equation 11:
The electric force gradient has a steeper dependence on Z than the electric force. It also has a steeper dependence on X and Y.
