The S-state model: a work horse for HRTEM

The S-state model describes the dynamical scattering of electrons in a specimen foil, consisting of atom columns parallel to the beam direction, such as a crystal or a particular crystal defect. In this model the electrons are considered to be trapped in the electrostatic potential of an atom column, in which it scatters dynamically. This picture allows physical insight, and it explains why a one-to-one correspondence is maintained between the exit wave and the projected structure, even in case of strong dynamical scattering. Furthermore the model can be parameterised in a simple closed analytical form. Apart from the computational advantages, the S-state model proves to be very useful to deduce the projected structure directly from the exit wave, so as to "invert" the dynamical scattering. In this paper the validity of the S-state model, is evaluated in much depth by a proper quantum mechanical treatment. The analytical parameterisation of the 1S eigenfunction and eigenenergy is discussed. It is shown that the method, even in case of small tilts, is valid for most thicknesses, currently used in HRTEM studies. Even for closely spaced atom columns, such as the dumbbells in Si [1 1 0], Sn [1 1 0] and GaN [1 1 0], the positions of the atom columns can be deduced with an accuracy of a few pm.