Additional theoretical information - Field Electron Emission (Field Emission, Tunnel

Developing the approximate CFE theory for metals given above is comparatively simple, for the following reasons: (1) Sommerfeld's free-electron theory, with its particular assumptions about the energy distribution of internal electron states, is adequately applicable to many metals as a first approximation. (2) In most cases metals have no surface states and (in many cases) the 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 sits mainly on the outer side of the top atomic layer, and no significant "field penetration" occurs. (4) Metals have high electrical conductivity : no significant voltage drops occur inside metallic emitters: this means that there are no factors impeding the supply of electrons to the emitting surface, and that the electrons in this region can be in effective local thermodynamic equilibrium and also in effective thermodynamic equilibrium with the electrons in the metallic support structure on which the emitter is mounted. (5) Atomic-level effects are ignored.

The development of "simple" theories of field electron emission, and in particular of Fowler–Nordheim-type equations, rests 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 not hold. For example, crystalline semiconductors do not have a free-electron-like band structure, have surface states, are subject to field penetration and band bending , and can exhibit both internal voltage drops and a statistical separation of the distribution of surface-state electrons from the distribution of electrons in the surface region of the bulk band structure (this separation is known as the "Modinos effect"). ^{[ 33 ] [ 92 ]}

In practice, the theory of the actual Fowler–Nordheim tunnelling process is largely the same for all materials (although the details of the barrier shape may differ, and for initial states that are localized rather than travelling-wave in form, a modified theory must be developed). However, despite such differences, it is expected (for situations of thermodynamic equilibrium) that all CFE equations will have exponents that behave in a broadly similar way. This is why applying Fowler–Nordheim-type equations to materials beyond the scope of the derivations given here often proves effective. If the only quantities of interest (such as the field enhancement factor) are those related to the slope of Fowler–Nordheim or Millikan–Lauritsen plots and to the exponent of the CFE equation, then Fowler–Nordheim-type theory often gives reasonable estimates. However, attempts to derive meaningful values of current density usually or always fail.

It should be noted that a straight line on a Fowler–Nordheim or Millikan–Lauritsen plot does not indicate that the emission from the material in question obeys a Fowler–Nordheim-type equation: it indicates only that the emission mechanism of individual electrons is probably Fowler–Nordheim tunnelling.

Different materials can have radically different energy distributions of their internal electron states, so the process of integrating current-density contributions over the internal electron states can lead to markedly different expressions for the pre-exponential functions of current density for different classes of materials. In particular, the power of the barrier field appearing in the pre-exponential function may differ from the original Fowler–Nordheim value of "2". Investigating effects of this kind is an active research topic. "Resonance" and "scattering " effects at the atomic level , if they occur, would also modify the theory.

Where materials are subject to field penetration and band bending, a necessary prerequisite is to have good theories of such effects (for each class of materials) before detailed CFE theories can be developed. In cases where voltage-drop effects arise, the theory of emission current may to a greater or lesser extent turn into a theory that includes internal transport effects, and become very complex.

• field emission microscope
• Field emission probes
• Field emitter array
• field emission display
• Franz–Keldysh effect