Field Electron Emission - Further Theoretical Information

Further Theoretical Information

Developing the approximate theory of CFE from metals above is comparatively easy, for the following reasons. (1) Sommerfeld's free-electron theory, with its particular assumptions about the distribution of internal electron states in energy, applies adequately to many metals as a first approximation. (2) Most of the time, metals have no surface states and (in many cases) metal wave-functions have no significant "surface resonances". (3) Metals have a high density of states at the Fermi level, so the charge that generates/screens external electric fields lies mainly on the outside of the top atomic layer, and no meaningful "field penetration" occurs. (4) Metals have high electrical conductivity: no significant voltage drops occur inside metal emitters: this means that there are no factors obstructing the supply of electrons to the emitting surface, and that the electrons in this region can be both in effective local thermodynamic equilibrium and in effective thermodynamic equilibrium with the electrons in the metal support structure on which the emitter is mounted. (5) Atomic-level effects are disregarded.

The development of "simple" theories of field electron emission, and in particular the development of Fowler-Nordheim-type equations, relies on all five of the above factors being true. For materials other than metals (and for atomically sharp metal emitters) one or more of the above factors will be untrue. For example, crystalline semiconductors do not have a free-electron-like band-structure, do have surface states, are subject to field penetration and band bending, and may exhibit both internal voltage drops and statistical decoupling of the surface-state electron distribution from the electron distribution in the surface region of the bulk band-structure (this decoupling is known as "the Modinos effect").

This, basically, is why this article is confined to the theory of CFE from bulk metals. The complications involved in presenting CFE theory for non-metals are too great to be dealt with satisfactorily via Wikipedia. In any case, the basic theory of CFE from bulk metals needs to be understood first.

In practice, the theory of the actual Fowler-Nordheim tunneling process is much the same for all materials (though details of barrier shape may vary, and modified theory has to be developed for initial states that are localized rather than are travelling-wave-like). However, notwithstanding such differences, one expects (for thermodynamic equilibrium situations) that all CFE equations will have exponents that behave in a generally similar manner. This is why applying Fowler-Nordheim-type equations to materials outside the scope of the derivations given here often works. If interest is only in parameters (such as field enhancement factor) that relate to the slope of Fowler-Nordheim or Millikan-Lauritsen plots and to the exponent of the CFE equation, then Fowler-Nordheim-type theory will often give sensible estimates. However, attempts to derive meaningful current density values will usually or always fail.

Note that a straight line in a Fowler-Nordheim or Millikan-Lauritsen plot does not indicate that emission from the corresponding material obeys a Fowler-Nordheim-type equation: it indicates only that the emission mechanism for individual electrons is probably Fowler-Nordheim tunneling.

Different materials may have radically different distributions in energy of their internal electron states, so the process of integrating current-density contributions over the internal electron states may give rise to significantly different expressions for the current-density pre-exponentials, for different classes of material. In particular, the power of barrier field appearing in the pre-exponential may be different from the original Fowler-Nordheim value "2". Investigation of effects of this kind is an active research topic. Atomic-level "resonance" and "scattering" effects, if they occur, will also modify the theory.

Where materials are subject to field penetration and band bending, a necessary preliminary is to have good theories of such effects (for each different class of material) before detailed theories of CFE can be developed. Where voltage-drop effects occur, then the theory of the emission current may, to a greater or lesser extent, become theory that involves internal transport effects, and may become very complex.