# Specifying the geometry¶

## Partitioning the system¶

In the simplest case, transport calculations can be performed on an atomistic structure comprising of two semi-infinite contacts. These contacts are usually one-dimensional periodic systems (wire-like) and connect to a device region.

In order to carry out a transport calculation with DFTB+, the geometry of the system must be carefully partitioned by the user and the structure must contain:

1. The extended molecule (the central region),
2. Two Principal Layers (see below) for the first contact,
3. Two Principal Layers of the second contact.

(additional contacts can also be included, but we will first start with the most common situation of a 2 contact geometry).

Figure 21 Partitioning of a simple system into an extended molecule and perfect contacts.

The extended molecule contains the atomistic device itself plus those parts of the attached contacts, which are directly influenced by the presence of the device (we can call them surfaces). Each contact, on the other hand, contains those parts of the actual contacts, which are far enough from the device not to be affected by it (the examples below will better clarify this point).

### Partitioning the system and contacts¶

The most important concept to bear in mind is that of principal layers (PLs). These are defined as contiguous groups of atoms that have interaction only with atoms of the immediately adjacent PLs. In practice these layers must contain a sufficient number of atoms in order to ensure that the hamiltonian and overlap interactions vanish before reaching the second nearest neighbour PLs (or can be considered negligible). The subdivision of the system into PLs is essential for the definition of the two contacts and becomes useful in the computation of all Green’s functions that exploit a recursive algorithm (See [PPD2008]).

Figure 22 Subdivision of a general structure into principal layers (PL), two for each contact and an arbitrary number within the extended molecule region.

As shown in Figure 22, the extended device molecule can contain an arbitrary number of PLs (>=1), but the layers themselves must follow a sequential ordering. The ordering of the PLs follow directly from the spatial ordering of the atoms in the DFTB+ structure.

Typically it is convenient to create the structure and then sort the atom along the transport direction before then partitioning up the system. In other cases, for example when nanowires are constructed, it is more convenient to repeat a PL unit for the desired length of the extended molecule.

The (perfect) contacts are defined by two principal layers. Unlike the layers of the extended molecule region, the PLs defining the contacts must follow additional rules:

1. The two principal layers of a given contact must be identical.
2. They must be rigidly shifted images of each other.
3. The first PL must be the closest to the device region.

### Ordering the atoms in the system¶

In order to ensure that the above prescriptions are met the numbering of the atoms must follow a precise ordering.

The atoms of the central region must be specified first in the structure, i.e. before the atoms of the contacts (see Partitioning the system and contacts for details). The atoms of each of the contacts then follow those of the central region in turn. The atoms of each contact must be grouped together sequentially (You specify either all atoms of the left contact first, and then those of the right one, or vice versa). The numbering field for the atoms in the gen format input is ignored (see Geometry for details), it is the order of the atoms in the structure that matters. We often find it useful for readability to number atoms in the device (or its principal layers) sequentially, then restart the count for each contact.

The numbering of the atoms within the first principal layer of a contact is arbitrary, but the same ordering of atoms must be applied to the second PL of that contact. Thus, the order that the atoms are listed in each of the two contact PLs must be the same and their location in the list must differ by the same amount for all atoms in the two PLS. The coordinates of the equivalent atoms in the two layers must also differ by the same vector (the contact vector) which translates all atoms of the first PL onto their equivalents in the second PL. These prescriptions are checked by DFTB+ and an error message is issued if the two PLs do not conform to these requirements. It is important to remember to number the atoms of the first PL to be the layer closer to the device, i.e. they must contain the atoms with the lowest indices in the contact.

Figure 23 Example for numbering of the atoms. Those in the device have the lowest indices, followed by the atoms of each contact, respectively. The two principal layers making up each contacts are shifted copies of each other (both in space and order of numbering).

### Supercell structures¶

DFTB+ can compute transport for structures that have periodicity in one or both directions that are transverse with respect to the transport direction. In this case the structure must be defined as a supercell and the rules listed above apply to the resulting cell. The real-space Poisson solver of DFTB+ limits the supercell lattices to being of orthorombic type (orthogonal vectors parallel to the cartesian axes, i.e. all angles between supercell vectors must be 90 degrees). In fact the supercell is always defined as being 3-dimensional, and the user should ensure that the dummy lattice vector along the transport direction is long enough to avoid superpositions between atoms in images along that direction.

The current version of DFTB+ only supports one supercell definition for the entire system, including central region and contacts. It may be redundant to observe that in this case the two contacts must be of the same periodicity in the directions transverse to the transport direction.

### Transport Block¶

The geometry must be defined in the transport block, as specified in the following example:

Transport {
Device {
# Device is the 1st to 24th atom in the geometry list
AtomRange = 1 24
}
Contact {
Id = "source"
# This contact starts at atom 25
AtomRange = 25 44
}
Contact {
Id = "drain"
# This contact starts at atom 45
AtomRange = 45 58
}
}