# Kelvin probe force microscope

File:Kpfm.png
In Kelvin probe force microscopy, a conducting cantilever is scanned over a surface at a constant height in order to map the work function of the surface.

Kelvin probe force microscopy (KPFM), also known as surface potential microscopy, is a noncontact variant of atomic force microscopy (AFM) that was invented in 1991. With KPFM, the work function of surfaces can be observed at atomic or molecular scales. The work function relates to many surface phenomena, including catalytic activity, reconstruction of surfaces, doping and band-bending of semiconductors, charge trapping in dielectrics and corrosion. The map of the work function produced by KPFM gives information about the composition and electronic state of the local structures on the surface of a solid.

KPFM is a scanned probe method where the potential offset between a probe tip and a surface can be measured using the same principle as a macroscopic Kelvin probe. The cantilever in the AFM is a reference electrode that forms a capacitor with the surface, over which it is scanned laterally at a constant separation. The cantilever is not piezoelectrically driven at its mechanical resonance frequency ${\displaystyle \omega _{0}}$ as in normal AFM although an alternating current (AC) voltage is applied at this frequency.

When there is a direct-current (DC) potential difference between the tip and the surface, the AC+DC voltage offset will cause the cantilever to vibrate. The origin of the force can be understood by considering that the energy of the capacitor formed by the cantilever and the surface is

${\displaystyle E={\frac {1}{2}}C[V_{dc}+V_{ac}sin(\omega _{0}t)]^{2}={\frac {1}{2}}C[2V_{dc}V_{ac}sin(\omega _{0}t)-{\frac {1}{2}}V_{ac}^{2}cos(2\omega _{0}t)]}$

plus terms at dc. Only the cross-term proportional to the ${\displaystyle V_{dc}\cdot V_{ac}}$ product is at the resonance frequency ${\displaystyle \omega _{0}}$. The resulting vibration of the cantilever is detected using usual scanned-probe microscopy methods (typically involving a diode laser and a four-quadrant detector). A null circuit is used to drive the DC potential of the tip to a value which minimizes the vibration. A map of this nulling DC potential versus the lateral position coordinate therefore produces an image of the work function of the surface.

A related technique, electrostatic force microscopy (EFM), directly measures the force produced on a charged tip by the electric field emanating from the surface. EFM operates much like magnetic force microscopy in that the frequency shift or amplitude change of the cantilever oscillation is used to detect the electric field. However, EFM is much more sensitive to topographic artifacts than KFPM and has not proven as useful. Both EFM and KPFM require the use of conductive cantilevers, typically metal-coated silicon or silicon nitride.

## Working Principle

The Kelvin probe force microscope or Kelvin force microscope (KFM) is based on an AFM set-up and the determination of the work function is based on the measurement of the electrostatic forces between the small AFM tip and the sample. The conducting tip and the sample are characterised by (in general) different work functions. When both elements are brought in contact, a net electric current will flow between them until the Fermi levels are aligned. The potential is called the contact potential (difference) denoted generally with ${\displaystyle V_{CPD}}$. An electrostatic force between tip and sample builds up, resulting from the net charge transfer. For the measurement a voltage is applied between tip and sample, consisting of a DC-bias ${\displaystyle V_{DC}}$ and an AC-voltage ${\displaystyle V_{AC}=\sin(\omega _{2}t)}$ of frequency ${\displaystyle \omega _{2}}$ at the second resonance frequency of the AFM cantilever

${\displaystyle V=(V_{DC}-V_{CPD})+V_{AC}\sin(\omega _{2}t)}$

Tuning the AC-frequency to the second resonance frequency of the cantilever results in an improved sensitivity and allows the independent and simultaneous imaging of topography and the contact potential. As a result of these biasing conditions, an oscillating electrostatic force appears, inducing an additional oscillation of the cantilever with the characteristic frequency ${\displaystyle \omega _{2}}$. The general expression of such electrostatic force not considering coulomb forces due to charges can be written as

${\displaystyle F={\frac {1}{2}}{\frac {dC}{dz}}V^{2}}$

The electrostatic force can be split up into three contributions, as the total electrostatic force F acting on the tip has spectral components at the frequencies ${\displaystyle \omega _{2}}$ and ${\displaystyle 2\omega _{2}}$.

${\displaystyle F=F_{DC}+F_{\omega _{2}}+F_{2\omega _{2}}}$

The DC component, ${\displaystyle F_{DC}}$, contributes to the topographical signal, the term ${\displaystyle F_{\omega _{2}}}$ at the characteristic frequency ${\displaystyle \omega _{2}}$ is used to measure the contact potential and the contribution ${\displaystyle F_{2\omega _{2}}}$ can be used for capacitance microscopy.

${\displaystyle F_{DC}=-{\frac {dC}{dz}}[{\frac {1}{2}}(V_{DC}-V_{CPD})^{2}+{\frac {1}{4}}V_{AC}^{2}]}$

${\displaystyle F_{\omega _{2}}=-{\frac {dC}{dz}}[V_{DC}-V_{CPD}]V_{AC}\sin(\omega _{2}t)}$

${\displaystyle F_{2\omega _{2}}=+{\frac {1}{4}}{\frac {dC}{dz}}V_{AC}^{2}\cos(2\omega _{2}t)}$

For contact potential measurements a lock-in amplifier is used to detect the cantilever oscillation at ${\displaystyle \omega _{2}}$. During the scan ${\displaystyle V_{DC}}$ will be adjusted so that the electrostatic forces between the tip and the sample become zero and thus the oscillation amplitude of the cantilever at the frequency ${\displaystyle \omega _{2}}$ becomes zero. Since the electrostatic force at ${\displaystyle \omega _{2}}$ depends on ${\displaystyle V_{DC}-V_{CPD}}$, ${\displaystyle V_{DC}}$ corresponds to the contact potential. Absolute values of the sample work function can be obtained if the tip is first calibrated against a reference sample of known work function. Apart from this, one can use the normal topographic scan methods at the resonance frequency ${\displaystyle \omega }$ independently of the above. Thus, in one scan, the topography and the contact potential of the sample are determined simultaneously.