The tensor-LEED approach

This section provides a rudimentary introduction to the tensor-LEED approach employed by TensErLEED and, consequently, also ViPErLEED. Note that this is neither, nor aims to be a comprehensive or rigorous introduction to the topic. The descriptions below are only intended to provide a quick overview of the method, and serve as explanation and motivation for the various sections of a LEED-I(V) calculation in ViPErLEED.

An in-depth description of all parts of the tensor-LEED approach is presented in Refs. 7, 8. The original publication of the TensErLEED software explains how the tensor-LEED approach is implemented in TensErLEED [6].

Reference calculation

The reference calculation determines the full-dynamic scattering (i.e., including all multiple-scattering contributions) of an electron wave incident at a “reference” structure. This calculation yields the complex scattering amplitudes \(A^{\mathrm{ref}}_{\mathbf{g}}\) (and intensities, via \(I = |A|^2\)) of all diffracted beams \(\mathbf{g}\) that are of interest for the requested structure.

As discussed in more detailed elsewhere [3, 5], the full-dynamic calculation is computationally demanding. However, it is possible to obtain accurate diffraction amplitudes for small deviations from the reference structure by using a first-order-perturbation approach [7]. The deviations from the reference structure may be geometric (i.e., altered atom positions), changes to atomic vibrational amplitudes, or chemical substitutions [9].

Each atom \(i\) is assigned a “\(t\) matrix”, \(t_i\), based on electron-scattering phase shifts and positions within the unit cell. The perturbed structure is consequently characterized by altered atomic \(t\) matrices, \(\tilde{t_i} = t_i + \delta t_i\).

The amplitude for a beam \(\mathbf{g}\) diffracted at the perturbed structure can be written as

\[A^{\mathrm{per}}_{\mathbf{g}} = A^{\mathrm{ref}}_{\mathbf{g}} + \sum_{i} \delta{}A_{i,\mathbf{g}} ,\]

that is, as the reference amplitudes plus a sum of delta amplitudes for the altered atoms. These delta amplitudes can be expressed as

\[\delta{}A_{i,\mathbf{g}} = \sum_{l,m;l',m'} T^{\mathrm{ref}}_{i,\mathbf{g};l,m;l',m'} \braket{\mathbf{r}_i,l,m| \delta{}t_{i} |\mathbf{r}_i,l',m'}\]

using the perturbations of the atomic \(t\) matrices, \(\delta t_i\), the tensorial quantities \(T^{\mathrm{ref}}_{i,\mathbf{g};l,m;l',m'}\), and the unperturbed positions of the atoms \(\mathbf{r}_i\). The sum runs over two sets of angular-momentum (\(l\), \(l'\)) and magnetic (\(m\), \(m'\)) quantum numbers. For a rigorous derivation, refer to the original work by Rous et al. [7] and to the TensErLEED paper by Blum and Heinz [6].

The quantities \(T^{\mathrm{ref}}_{i,\mathbf{g};l,m;l',m'}\) only depend on the reference structure and are commonly just referred to as “tensors”. Importantly, the tensors can be calculated during the reference calculation. They are the starting point for the subsequent delta amplitude calculation and structure search.

Delta-amplitude calculation

The individual perturbations to the reference structure may be combinations of geometric displacements, changes in the vibrational amplitudes, or chemical substitutions. As tensor LEED is based on first-order perturbation theory, these perturbations — and the resulting amplitude changes — can be treated on an atom-by-atom basis.

For each atom \(i\) and for each requested perturbation \(p\) to that atom, the delta-amplitude calculation evaluates the perturbed \(t\) matrix \(\tilde{t}_{i,p} = t_i + \delta t_{i,p}\) and the corresponding amplitude changes

\[\delta{}A_{i,\mathbf{g},p} = \sum_{l,m;l',m'} T^{\mathrm{ref}}_{i,\mathbf{g};l,m;l',m'} \braket{\mathbf{r}_i,l,m| \delta t_{i,p} |\mathbf{r}_i,l',m'} .\]

The resulting delta-amplitudes are used in the structure search to calculate the perturbed intensities for each structure candidate [6].

Note

Depending on the size of the unit cell and the requested perturbations, the parameter space may become very big.

Tensor-LEED errors

Since the tensor-LEED approach is based on first-order perturbation theory, it is inherently limited to small perturbations. The larger the perturbation, the larger the error incurred by the approximation and the less reliable the result.

This should be kept in mind when interpreting the results of any ViPErLEED segment that uses the tensor-LEED approach (i.e., the structure search and the error calculation). In particular, it is strongly recommended to run a new reference calculation after the structure optimization has converged to get rid of any accumulated errors. It is also usually necessary to iterate between structure search and reference calculation to obtain the best possible fit.

There is one case, however, in which a full-dynamic calculation can yield more erroneous results than tensor LEED. The full-dynamic reference calculation cannot provide exact results when an atom has mixed chemical composition and the elements have different optimized positions. This is because only one position can be specified for each atom. In this case, the tensor-LEED approximation is the only viable alternative. It should anyway be used with care. In particular, the position deviations of the different chemical species from the “mean” position used for the full-dynamic calculation should be small.