Electron channelling based crystallography.

Electron channelling occurs when the incident electron beam is parallel to the atom columns of an object, such as a crystal or a particular crystal defect. Then, the electrons are trapped in the electrostatic potential of an atom column in which they scatter dynamically. This picture provides physical insight and explains why a one-to-one correspondence is maintained between the exit wave and the projected structure, even in case of strong dynamical scattering. Moreover, the theory is very useful to invert the dynamical scattering, that is, to derive the projected structure from the exit wave. Finally, it can be used to determine the composition of an atom column with single atom sensitivity or to explain dynamical electron diffraction effects. In this paper, an overview of the channelling theory will be given together with some recent applications.

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.

Does a monochromator improve the precision in quantitative HRTEM?

This paper addresses the question as to what extent the incorporation of a monochromator in an electron microscope can enhance the performance of high resolution transmission electron microscopy (HRTEM). The monochromator will reduce the chromatic aberration, and hence the information limit, at the expense of beam current, leading to a decrease in signal intensity and a corresponding decrease in signal-to-noise ratio (SNR). Both aspects, information limit and SNR, have been included in a quantitative evaluation based on the statistical precision with which the position of an atom column can be estimated. It is shown that the effect of a monochromator on the attainable precision depends on the microscope and monochromator parameters, as well as on the characteristics of the object.

About forbidden and weak reflections.

Reflections forbidden under the single scattering approximation are not expected to remain extinct for crystal thicknesses for which multiple scattering becomes important, except for reflections of the Gjönnes-Moodie type. However, it has been observed that in many crystals with the incident beam along a zone axis, such as diamond-like crystals along the [110] zone, reflections forbidden under the single scattering approximation remain very weak up to large thicknesses. This is hard to explain in terms of many-beam dynamical scattering in Fourier space. The picture becomes clear if one describes the scattering in real space in terms of the channelling of the electrons along the atom columns parallel to the zone axis. In that case the exit wave of each atom column can be described by the S-state model, which is radially symmetric around the centre of the atom column. As a consequence the exit wave shows the same symmetry as the projected potential, so that the reflections forbidden under the single scattering approximation remain extinct. This condition only breaks down when the crystal thickness becomes so large that the S-state model becomes invalid, which is a function of the distance between neighbouring atom columns and/or the tilt from the exact zone axis. The sensitivity for small tilts is also in agreement with very old observations that have not been explained thus far.