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This question goes into the very nature of the work function that the Kelvin Probe Force Microscope (KPFM) measures.

https://en.wikipedia.org/wiki/Kelvin_probe_force_microscope

Let's say, you have a battery of 1.5 V, you connect two identical metal leads to both ends. If you measure the work function on both ends with a KPFM, would they be the same or would the work function be different by 1.5 eV?

A co-worker and I were discussing this, and unfortunately, we don't have a KPFM to test this. The idea is that since the electrons have energies differing by 1.5 eV, there would be a difference in the workfunction between the two metals by 1.5 eV. I disagree with this, but cannot give a good argument why I think the workfunction for both leads would be the same.

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KPFM can detect if a voltage is applied to a metal, as it measures the contact potential difference due to the work function diffence between the probe tip material and the metal surface. So if there is a voltage applied to the metal (sample), you are then conceptually changing the fermi level of the metal hence changing the work function difference between the sample and probe tip.

However, in your thought experiment there is a small detail that you have to consider. For KPFM to work, the probe tip and the sample metal have to be externally connected. So whether or not you will measure the applied voltage due to the external battery you added depends on how you connect the setup.

In the assumed case where each metal lead is connected to the terminals of a 1.5V battery in an open circuit, during measurement each metal lead will be connected to the probe tip in the setup. In that case you will measure the same value for both metal leads and the battery will have no effect.

The other case where you will detect the applied voltage on one of the metal leads would be if the battery is connected to the probe tip within the setup (a series voltage source to the compensating kelvin voltage). So a practical connection would be that where one of the metal leads is connected to the battery with the other lead grounded with the same ground of the kelvin probe tip. The resulting measurement would be the contact potential diffence between the probe tip and the metal lead + the 1.5 eV due to the battery.

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