Practical Electron Microscopy and Database

The beam energy in EMs is significantly broadened by the interaction of the electrons with the specimen.

For a field emission source, the beam of electrons emitted presents an energy profile described by the Fowler-Nordheim (F-N) Distribution [1],

          ------------- [4240a]

          Δ = µ - E ---------------------- [4240b]

          b = 6.8 x 10⁷α ---------------------- [4240c]

where,
          µ -- The chemical potential (Typical value ~ 11 eV),
          φ -- The work function (for a tungsten tip as 4.5 eV),
          α -- The image correction term (Typical value ~ 0.85),
          F -- The electric field strength (Typical value ~ 3 x 10⁷ V/m),
          E -- The accelerating voltage on the incident electrons.

The F-N energy distribution of emitted electrons includes two distinct regions [2]: a low energy Fermi tail and a high energy tunnelling tail. The low energy Fermi tail is a property of the Fermi surface of the tip material (e.g. tungesten) and is independent of the extraction voltage. The slope of the high energy tunnelling tail is determined by the field strength. Those tails cause beam energy-broadening in field emission guns.

[1] Fowler, R.H., Nordheim, L. Electron emission in intense electric fields. Proc. R. Soc. A 119, 173-181 (1928).
[2] Kimoto, K. et al. 0.23eV energy resolution obtained using a cold field-emission gun and a streak imaging technique. Micron 36, 465-469 (2005).