Abstract
We already used the term work function when we introduced the tunneling barrier height in STM. The work function can be considered as the energy difference between the vacuum level and the Fermi level of a metal. Here we will see that also a surface term contributes to the work function. The work function is a measurable quantity and the operative definition of the work function is that it is the energy required to remove an electron from the bulk Fermi level of a metal to a certain distance from the solid.
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Notes
- 1.
This distance is specific to the actual type of measurement performed.
- 2.
This can be seen from a simple 1D particle in a box model, where the energy of an electron state as a function of the quantum number \(n\) and size of the box \(L\) is
$$\begin{aligned} E(L) = \frac{\hbar ^2 \pi ^2 n^2}{2 m_{\mathrm {e}} L^2}. \end{aligned}$$(9.3)With increasing L (“spill out” of charge) the energy decreases.
- 3.
In classical electrostatics it is shown that the force between an electron at distance \(d\) from a conducting plate is the same as the force between the electron and a positive elementary charge located at a distance \(2d\) from the electron (image charge), i.e. \( - e^2/(4 \pi \varepsilon _0 2 d)\). Integrating the negative of this force from infinity to \(d\) results in the (image) potential of the electron (relative to a position at infinity) as
$$\begin{aligned} V_{\mathrm {image}}(d) = \int \limits _{\infty }^d{\frac{e^2}{4 \pi \varepsilon _0 2 r} dr} = \frac{-e^2}{4 \pi \varepsilon _0} \frac{1}{4 d}. \end{aligned}$$(9.5).
- 4.
All net charges are located at the surface of a metal, since the electric field vanishes in the interior of a metal.
- 5.
It is always assumed that the electron is at rest, i.e. there is no kinetic energy contribution to the work.
- 6.
We assume semi infinite crystals so that no surface charges are present and thus no electric fields occur outside the crystals. Since in Fig. 9.3a macroscopic distance between both metals is assumed, the work function rises within 100 nm quasi vertically to \(E_{\mathrm {vac}} = E_{\mathrm {vac}}^\infty \).
- 7.
This is done since the force (not the current) is measured in a scanning force microscopy setup.
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Voigtländer, B. (2015). Work Function, Contact Potential, and Kelvin Probe Scanning Force Microscopy. In: Scanning Probe Microscopy. NanoScience and Technology. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45240-0_9
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DOI: https://doi.org/10.1007/978-3-662-45240-0_9
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