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.

Structure search

Once the amplitude changes for all required perturbations have been obtained, the final diffraction amplitudes can be calculated using a simple superposition: the overall amplitude change is the sum

\[\delta{}A^\mathrm{tot}_\mathbf{g} = \sum_{i,p} \delta{}A_{i,\mathbf{g},p}\]

of the amplitude changes for all the affected atoms. The total diffracted amplitudes then result by adding the amplitude changes to the reference amplitudes.

This way, scattered intensities can be obtained for any structure in the configuration space. Different candidate structures in this configuration space are compared to experimental data via the \(R\) factor. The best-fit structure results from a minimization of the \(R\) factor in the configuration space.

While conceptually simple, this optimization is practically very challenging, and usually constitutes the computationally most expensive part of a LEED-I(V) calculation. By using the tensor-LEED approach, the problem is tractable, even for systems with relatively large unit cells. Running a full-dynamic calculation for every configuration is orders of magnitude more computationally expensive [7].

There are some fundamental caveats concerning the structure optimization in the tensor-LEED approximation:

Since the tensor-LEED method is perturbative, it only works reliably for small perturbations. What exactly constitutes a small perturbation depends on the system. The applicability of tensor LEED is normally limited to displacements of at most 0.2 Å (only in very simple cases up to 0.3 Å) [7]. For larger displacements, the optimization might give the right trends but the results should be taken with caution (see also Tensor-LEED errors).

When the trajectory of the structure optimization approaches the limit of applicability of tensor LEED, the range of the structural search can be extended by running new reference and delta-amplitudes calculations.
The parameter space grows quickly with the number of atoms in the unit cell. Luckily, many symmetries inherent to the surface structure can be exploited to eliminate redundant parameters. Specifically, displacements of symmetry-linked atoms must always happen in a concerted fashion. If that were not the case, the symmetry would be broken and a different LEED pattern would result.

To make use of these symmetries and the resulting reduction of the parameter space, it is necessary to know and constrain the surface symmetry of the slab. While manually finding out the surface symmetry is normally an easy task, maintaining it is not. This is especially true for geometric displacements: It would require manually defining matching displacement vectors for all symmetry-linked atoms.

Fortunately for the user, automatic symmetry detection and constraint is one of the main features of ViPErLEED.
When using Pendry’s \(R\) factor, the \(R\) factor hypersurfaces are inherently rough [10]. Users should be aware that local minima are possible and that the optimization algorithm might get trapped in these minima if the parameter space is not opened up sufficiently. Simultaneous optimization over too many parameters is also a common cause of trapping in local minima.
As described above, the tensor-LEED implementation in TensErLEED separates the calculation of delta amplitudes and the structure optimization into two independent stages. As a direct consequence, the optimization can only be performed on a predefined grid of perturbations. Further, to achieve the best possible fit, the grid-based nature makes it necessary to run multiple sets of delta-amplitude calculations and structure optimizations with increasingly finer pitch.

Note

Starting with a fine grid over a large variation range is not recommended: Too many grid points per parameter significantly slow down the convergence.
The structure search implemented in TensErLEED has the additional limitation that geometric displacements are limited to one dimension per atom. During each search run, atoms can only be displaced along a predefined curve rather than freely in 3D space. To optimize the position of atoms in three dimensions, multiple sequential search runs are needed. See the entry on the DISPLACEMENTS file for details and workarounds (such as looping searches).

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.