Theory and Experimental Set-Up of a Scanning Kelvin Probe

For a parallel plate capacitor with the sinusoidally modulated plate separation d(t) given by Eq. (2) (do=average distance between the parallel plates. Ad=vibration amplitude and ( =angular frequency of vibration), the time dependent capacitance C(t) is given by Eq. (3) with C(t) defined by Eq. (3). 

Assuming that the average distance between the parallel plates is much greater than the vibration amplitude (Ad do), this term can be simplified by using (do-i-Ad cos cot) do. 

The amplitude I of the current i(t) could be measured by performing a demodulation with a lock-in amplifier (LIA) at the angular frequency co of the sinusoidal vibration of the needle, leading to Eq. (6). 

The measurement of the Volta potential difference is performed by zeroing the Kelvin current amplitude, which is done automatically with a feedback loop, and reading the compensation voltage. Under the conditions of Eq. (7) the external voltage, which is identical to the Volta potential difference, is given by the energy difference between the Eermi level of the sample and the Fermi level of the probe (which is known in many cases) 

The work function W of a metal is defined as the energy needed to move an electron from the Fermi level of the metal to a point at an infinite distance outside the surface (vacuum level). Thus for a metal surface under ultrahigh-vacu-um conditions, only the metal surface has to be passed for this transfer and therefore the work function is a well-defined quantity. 

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