Field-emission from high-index crystallographic faces: A scattering approach

We reconsider the usual theory of electron field-emission to deal with the case of high index crystallographic directions. The general requirement for an electron to be emitted is k_∥ + G ≈ 0, where k_∥ is the component of the electron wavevector parallel to the crystal surface, and G is a vector of the two-dimensional reciprocal lattice of the surface. In the case of high index crystallographic directions this requirement can be fulfilled for a large number of k points on the Fermi surface. A scattering approach is proposed to solve the problem of wavefunction matching at the crystal boundary. The calculation is carried out to the first Born approximation for the case of free electrons. Then the extension to arbitrary band structures in discussed, and a detailed treatment is given in the case of a nearly free electron model. The main result is that every k on the Fermi surface can contribute to the emitted current, and not only those with k_∥ ≈ 0 as in usual “specular” field-emission theory. This can yield considerable changes in such cases where there is no k vector normal to the crystal surface at the Fermi level.