Multislice - Practical Considerations

Practical Considerations

The basic premise is to calculate diffraction from each layer of atoms using Fast Fourier Transforms (FFT) and multiplying each by a phase grating term. The wave is then multiplied by a propagator, inverse Fourier Transformed, multiplied by a phase grating term yet again, and the process is repeated. The use of FFTs allows a significant computational advantage over the Bloch Wave method in particular, since the FFT algorithm involves steps compared to the diagonalization problem of the Bloch wave solution which scales as where is the number of atoms in the system. (See Table 1 for comparison of computational time).

The most important step in performing a multislice calculation is setting up the unit cell and determining an appropriate slice thickness. In general, the unit cell usd for simulating images will be different from the unit cell that defines the crystal structure of a particular material. The primary reason for this due to aliasing effects which occur due wraparound errors in FFT calculations. The requirement to add additional “padding” to the unit cell has earned the nomenclature “super cell” and the requirement to add these additional pixels to the basic unit cell comes at a computational price.

To illustrate the effect of choosing a slice thickness that is too thin, let us consider a simple example. The Fresnel propagator describes the propagation of electron waves in the z direction (the direction of the incident beam) in a solid:

Where is the reciprocal lattice coordinate, z is the depth in the sample, and lambda is the wavelength of the electron wave (related to the wave vector by the relation ). Figure shows vector diagram of the wave-fronts being diffracted by the atomic planes in the sample. In the case of the small-angle approximation ( 100 mRad) we can approximate the phase shift as . For 100 mRad the difference is fleeting so . For small angles this approximation holds regardless of how many slices there are, although choosing a greater than the lattice parameter (or half the lattice parameter in the case of perovskites) for a multislice simulation would be rather problematic.

Additional practical concerns are how to effectively include effects such as inelastic and diffuse scattering, quantized excitations (e.g. plasmons, phonons, excitons), etc. There was one code that took these things into consideration through a coherence function approach called Yet Another Multislice (YAMS), but the code is no longer available either for download or purchase.