Work function
Here "immediately" means that the final electron position is far from the surface on the atomic scale, but still too close to the solid to be influenced by ambient electric fields in the vacuum.
Rearranging the above equation, one has where V = −EF / e is the voltage of the material (as measured by a voltmeter, through an attached electrode), relative to an electrical ground that is defined as having zero Fermi level.
The observed data from these effects can be fitted to simplified theoretical models, allowing one to extract a value of the work function.
[9] Many techniques have been developed based on different physical effects to measure the electronic work function of a sample.
One may distinguish between two groups of experimental methods for work function measurements: absolute and relative.
The work function is important in the theory of thermionic emission, where thermal fluctuations provide enough energy to "evaporate" electrons out of a hot material (called the 'emitter') into the vacuum.
If these electrons are absorbed by another, cooler material (called the collector) then a measurable electric current will be observed.
Thermionic emission can be used to measure the work function of both the hot emitter and cold collector.
If the photon's energy is greater than the substance's work function, photoelectric emission occurs and the electron is liberated from the surface.
Similar to the thermionic case described above, the liberated electrons can be extracted into a collector and produce a detectable current, if an electric field is applied into the surface of the emitter.
Moreover, the minimum energy can be misleading in materials where there are no actual electron states at the Fermi level that are available for excitation.
For example, in a semiconductor the minimum photon energy would actually correspond to the valence band edge rather than work function.
The electric field can be varied by the voltage ΔVsp that is applied to the probe relative to the sample.
[14] Due to the complications described in the modelling section below, it is difficult to theoretically predict the work function with accuracy.
The work function is not simply dependent on the "internal vacuum level" inside the material (i.e., its average electrostatic potential), because of the formation of an atomic-scale electric double layer at the surface.
The reason for the dependence is that, typically, the vacuum level and the conduction band edge retain a fixed spacing independent of doping.
[17] If there is a large density of surface states, then the work function of the semiconductor will show a very weak dependence on doping or electric field.
One of the earliest successful models for metal work function trends was the jellium model,[19] which allowed for oscillations in electronic density nearby the abrupt surface (these are similar to Friedel oscillations) as well as the tail of electron density extending outside the surface.
This model showed why the density of conduction electrons (as represented by the Wigner–Seitz radius rs) is an important parameter in determining work function.
The jellium model is only a partial explanation, as its predictions still show significant deviation from real work functions.
More recent models have focused on including more accurate forms of electron exchange and correlation effects, as well as including the crystal face dependence (this requires the inclusion of the actual atomic lattice, something that is neglected in the jellium model).
A theoretical model for predicting the temperature dependence of the electron work function, developed by Rahemi et al. [21] explains the underlying mechanism and predicts this temperature dependence for various crystal structures via calculable and measurable parameters.
