This mini-tutorial, which is a bit different in nature compared to others, will explain how to properly design the electrodes in a two-probe system. Although this may appear to be a rather simple matter (pick a material and create a suitable surface cell) and people tend to focus more on the specifics of the scattering region - which indeed is where most of the physics happens - there are some subtle points that should be understood about the electrodes, not least in order to ensure the quality of the results.

The trickiest part is choosing a proper length in the transport direction of the electrode cell. For reasons to be explained below, if the electrode is too short we are effectively introducing an approximation in the calculation, which may influence both the accuracy of the results and the convergence of the self-consistent loop.

First, however, let us review some basic things about the electrodes.

Electrode unit cell

• The transport direction in ATK is always Z. Therefore, the electrode cell should have the C axis (the 3rd unit cell vector) parallel to Z.
• The two other electrode unit cell vectors (A and B) should be perpendicular to Z, i.e. they should span the XY plane.

These two criteria are always checked by ATK and if they are not fulfilled, the calculation will not proceed. For example, the smallest unit cell for an fcc [111] surface is hexagonal:

vector_a = [2.88902617589, 0.0, 0.0]*Angstrom
vector_b = [-1.44451308795, 2.50197006052, 0.0]*Angstrom
vector_c = [0.0, 0.0., 7.07663998448]*Angstrom
lattice = UnitCell(vector_a, vector_b, vector_c)

Periodicity and vacuum padding

• The electrode cell should be periodic in Z.
• The electrode cell should be periodic in X and Y (or quasi-periodic, in the case of 1D or 2D systems like atomic wires, nanotubes, or sheets).

The first of these requirements is the reason why, when we cleave an fcc system along [111], we obtain a surface cell with 3 layers, with the stacking sequence A-B-C. The picture below shows the result after cleaving silver in the Builder.

It is however not required that the left electrode is the same as the right electrode. That is, the entire two-probe system may be non-periodic, and in fact this is the case as soon as we apply a bias, even if the electrodes are geometrically identical. The absense of a periodicity requirement in the transport direction is precisely why ATK is able to compute the transport properties of a two-probe system at finite bias.

The second point above is related to the boundary conditions used when solving the Poisson equation. Even for a two-probe system computed with the multi-grid solver, the basic assumption is that the system is periodic in X and Y.

This is the reason why one should pad 1D and 2D systems with a large enough vacuum cell, such that first of all there is no basis set overlap between repeated atoms, and secondly no residual electrostatic interactions, although this is sometimes harder to achieve without making the cell very large, causing the calculation to be very expensive in memory and time. This point must also be considered for e.g. molecular junctions with 3D surface cells; not only must the cell often be repeated so that the molecule fits in the simulation cell, but there must be also be sufficient vacuum between the repeated copies of the molecules so that we can say that we truly model the transport properties of an isolated molecule.

The X/Y periodicity assumption can be very helpful, as it allows for calculations of layered interfaces with a rather small unit cell.

Electrode length in Z - background

From the requirements above, it seems clear that we should use 3 layers in the fcc [111] electrode (to make it periodic, and perpendicular), and this is what you see in most publications. We'll get back to this system below, but for now it is more instructive to consider a simpler system.

Consider the "standard" demonstration two-probe system, the Li-H2-Li chain (use e.g. in the basic transport tutorial for ATK, and the I-V curve mini-tutorial). Given the requirements stated above, a single atom should constitute a good enough electrode cell:

However, this calculation will not even converge! The reason for this is a mismatch between the Fermi level of the electrode and the central region that arises if the electrode is too short. In some other cases, the calculation may converge, but the results will not be correct, for the same reason. Such inaccuracies are hard to detect, which is why it is so important to understand and design the electrode properly from the beginning!

How can such a Fermi level mismatch appear, and how can we avoid it? To understand this, recall from above that in ATK we do not require any periodicity in the transport direction. To avoid treating an infinitely long system, as we basically would have to then, the two-probe method is ingenously constructed to treat the electrodes separately, as finite systems, after which they are coupled to the (also finite) central region. (For details on this, please refer to the methodology papers).

A specific part in this algorithm requires that 

• The atoms in the electrode unit cell should only interact with atoms in the nearest neighbor cell (in the Z direction).

In a sense it is not so much a requirement as an assumption: if there anyway are longer-reaching interactions, they will be ignored. This creates a mismatch between the two-probe system and the electrode Hamiltonians (which is computed as a truly periodic system, with all interactions included), and in result the electrode Fermi level will not match exactly the two-probe equillibrium Fermi level.

An essential element in this is, that the SIESTA basis set orbitals have a finite range, so the requirement can be strictly fulfilled for a finite electrode cell size; this would not be possible with Gaussian or plane-wave basis.

We should also note that there are two different types of "interactions" to be considered:

• Direct basis set overlaps (two-center terms in the Hamiltonian)
• Indirect interactions between atoms via pseudopotentials (three-center terms)

The direct interactions are crucial to include; neglecting any such overlaps constitutes are serious approximation that will compromise the results. The second-order interactions are of less importance, and in many cases their influence on the results can be safely neglected. They can however be important in systems where the density of states changes rapidly around the Fermi level, and where it therefore is critical to determine the Fermi level accurately.

In fact, it is easy to show that a sufficient criterion for including all possible interactions is:

• The electrode cell should be longer than twice the longest interaction radius.

If we are content with only including (for certain) all direct overlaps, then the relevant interaction radius is simply the basis set radius.

The criterion given above is actually often too strict (the Li-H2-Li chain is an example, as we will see very soon), but using the basis set radius as the interaction radius, it anyway serves as a good rule of thumb, since most of the second-order interactions will then be included as well.

Ensuring a proper electrode length in practice

We are still left with the all-important question: how do I know if my electrode is long enough? For that, we need to find out the longest interaction radius for a given system, and compare this to the actual distances between the atoms (this constitues a more accurate criterion).

The picture below illustrates this procedure for a chain of Li atoms. The image is to scale, assuming an interaction radius of 6.05 Å (the ATK-DFT basis set radius of Li is 4.04 Å, and the largest projector radius is 2.01 Å), a distance of a=2.9 Å between the Li atoms (the actual energetic minimum is closer to 3.0 Å, but because of the interaction radius is so close to 2a, using 2.9 Å illustrates the point better), and an electrode containing 3 atoms, giving a cell length of 8.7 Å.

This picture tells us that the electrode is too short to include all second-order interactions: the thicker yellow circle, centered on the 3rd atom in the original cell, intersects the thicker blue circle, centered on the first atom in the next-nearest neighbor cell. Since the basis set is quite a bit shorter, the cell is however long enough to include all basis set overlaps, and indeed the cell is longer than twice the basis set radius (about 8 Å).

On the other hand, it is clear that by extending the cell to 4 Li atoms, we would include all interaction, despite the fact that the cell is then still shorter (11.6 Å) than twice the total interaction radius (about 12 Å). The reason is that this criterion is based on the extreme situation where there is an atom very close to the right-hand edge of the cell, and another atom very close to the left-hand edge. This wouldn't really occur in a real physical system, and thus in practical cases, the criterion can usually be relaxed a bit.

To draw the system like above, and drawing circles (spheres in 3D!) around each atom to find out which ones are interacting, is obviously not practical for any realistic configuration. Instead, we here provide a tool that can be used in VNL to determine if the electrodes are long enough. To use it,

1. Download the script from here
2. Drop it on the Custom builder icon in the main VNL window
3. Drop your two-probe geometry on the drop-zone labeled "Drop system here"
4. If the system already has a calculator attached, all parameters from it will be used; otherwise you can select DFT or Hückel manually (all default parameters will apply)
5. In the log panel you will now see a report, stating the important lengths, and a final conclusion regarding the length of the electrode

The picture below shows the result when we use this tool to analyze the Li-H2-Li system used in the example above (with DFT).

The reported "trunction distance" is the smallest distance between atoms in the electrode and atoms in the next-nearest neighbor cell. Interactions between these atoms will be ignored, and thus if this distance is shorter than the maximum interaction range, the corresponding contribution to the Hamiltonian (either two- or three-center integral term) will not be included.