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# Electrostatic force microscope

Electrostatic force microscopy (EFM) is a type of dynamic non-contact atomic force microscopy where the electrostatic force is probed. ("Dynamic" here means that the cantilever is oscillating and does not make contact with the sample). This force arises due to the attraction or repulsion of separated charges. It is a long-ranged force and can be detected 100nm from the sample. For example, consider a conductive cantilever tip and sample which are separated a distance z usually by a vacuum. A bias voltage between tip and sample is applied by an external battery forming a capacitor, C, between the two. The capacitance of the system depends on the geometry of the tip and sample. The total energy stored in that capacitor is $U = - \frac{1}{2} C \Delta V^2$. The battery works to maintain a constant voltage, ΔV, between the capacitor plates (tip and sample). By definition, taking the negative gradient of the energy gives the force. The z component of the force (the force along the axis connecting the tip and sample) is thus:

$F_{electrostatic} = \frac{1}{2} \frac{\partial C}{\partial z} \Delta V^2$.

The electrostatic force can be probed by changing the voltage, and that force is parabolic with respect to the voltage. One note to make is that ΔV is not simply the voltage difference between the tip and sample. Since the tip and sample are often not the same material, and furthermore can be subject to trapped charges, debris, etc., there is a difference between the work functions of the two. This difference, when expressed in terms of a voltage, is called the contact potential difference, VCPD This causes the apex of the parabola to rest at ΔV = VtipVsampleVCPD = 0. Typically, the value of VCPD is on the order of a few hundred millivolts. Forces as small as piconewtons can routinely be detected with this method.

With an electrostatic force microscopes, like the atomic force microscope it is based on, the sample can be placed underwater.

## References

L. Kantorovich, A. Livshits, and M. Stoneham, J. Phys.:Condens. Matter 12, 795 (2000)

• "Electrostatic Force Microscope for Probing Surface Charges in Aqueous Solutions" by S. Xu and M.F. Arnsdorf