Work function

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The work function is the minimum energy (usually measured in electron volts) needed to remove an electron from a solid to a point immediately outside the solid surface. Here "immediately" means that the final electron position is far from the surface on the atomic scale but still close to the solid on the macroscopic scale. The work function is an important property of metals. The magnitude of the work function is usually about a half of the ionization energy of a free atom of the metal.

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[edit] Work Function and Surface Effect

Work function W of a metal is closely related to its Fermi energy level \epsilon_F \; yet the two quantities are not exactly the same. This is due to the surface effect of a real-world solid: a real-world solid is not infinitely extended with electrons and ions repeatedly filling every primitive cell over all Bravais lattice sites. Neither can one simply take a set of Bravais lattice sites \{R\} \; inside the geometrical region V which the solid occupies and then fill undistorted charge distribution basis into all primitive cells of \{R\} \;. Indeed, the charge distribution in those cells near the surface will be distorted significantly from that in a cell of an ideal infinite solid, resulting in an effective surface dipole distribution, or, sometimes both a surface dipole distribution and a surface charge distribution.

It can be proven that if we define work function as the minimum energy needed to remove an electron to a point immediately out of the solid, the effect of the surface charge distribution can be neglected, leaving only the surface dipole distribution. Let the potential energy difference across the surface due to effective surface dipole be W_S \;. And let \epsilon_F \; be the Fermi energy calculated for the finite solid without considering surface distortion effect, when taking the convention that the potential at r \rightarrow \infty \; is zero. Then, the correct formula for work function is:

W = - \epsilon_F +W_S \;

Where \epsilon_F \; is negative, which means that electrons are bound in the solid.

[edit] Example

For example, Caesium has ionization energy 3.9 eV and work function 1.9 eV.

[edit] Photoelectric work function

The work function is the minimum energy that must be given to an electron to liberate it from the surface of a particular metal. In the photoelectric effect if a photon with an energy greater than the work function is incident on a metal photoelectric emission occurs. Any excess energy is given to the electron as kinetic energy.

Photoelectric work function is

  • φ = hf0,

where h is the Planck's constant and f0 is the minimum (threshold) frequency of the photon required to produce photoelectric emission. When an electron gains energy, it jumps from one energy level to another in "quantum leaps". This process is called exciting an electron, and the higher energy levels are called "excited states" while the bottom level is called "ground state".

[edit] Thermionic work function

The work function is also important in the theory of thermionic emission. Here the electron gains its energy from heat rather than photons. In this case, as for an electron escaping from the heated negatively-charged filament of a vacuum tube, the work function may be called the thermionic work function. Tungsten is a very common metal for vacuum tube elements, with a work function of approximately 4.5 eV.

Thermionic emission requires a filament heating current (if), to maintain a temperature of 2000-2700 K. A saturation state of the filament current is reached, where a minor change in the filament current does not affect the beam current. The electron gun is then operated with the filament current very near the potential to overcome the work function (W)(Goldstein, 2003) The thermionic work function depends on the orientation of the crystal and will tend to be smaller for metals with an open lattice, larger for metals in which the atoms are closely packed. The range is about 1.5–6 eV. It is somewhat higher on dense crystal faces than open ones.

[edit] Applications

In electronics the work function is important for design of the metal-semiconductor junction in Schottky diodes and for design of vacuum tubes.

[edit] See also

  • free energy for the Helmholtz free energy equation, which is the thermodynamic work, note that this work is not related to electron emission and is thus not directly related to the work function.
  • Electron affinity. See NEA cathode for an application to condensed matter.

[edit] References

As a book:

  • Solid State Physics, by Ashcroft and Mermin. Thomson Learning, Inc, 1976
  • Goldstein, Newbury, et al, 2003. Scanning Electron Microscopy and X-Ray Microanalysis. Springer, New York.

For a quick reference to values of work function of the elements:

  • Herbert B. Michaelson, "The work function of the elements and its periodicity". J. Appl. Phys. 48, 4729 (1977)

[edit] External links

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