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Helical geometries

Helical geometries#

[Input: recipes/scripts/repeatGen.pl]

This section describes calculations for geometries with structures which can be described using helical (objective) boundary conditions.

Currently DFTB+ only supports non-SCC calculations with a combination of helical and Cn rotational symmetries.

A gen format for helical geometries looks something like:

2  H
C
1 1    0.2756230044E+01    0.2849950460E+01    0.1794011798E+01
2 1    0.2756230044E+01    0.2849950460E+01    0.3569110265E+00
0 0 0
0.2140932670E+01 18.0 10

This example is for a (10,0) carbon nanotube with a repeat distance along the tube of 2.141 Angstroms and the next layer of atoms is rotated by 18 degrees, with an additional C10 rotational symmetry. In this case the structure rotational repeats every 360 / 10 = 36 degrees, which is commensurate with twice the 18 degree helical twist.

Repeated nanotube a) along and b) across the tube axis

Figure 60 Repeated nanotube a) along and b) across the tube axis with the 2 atom objective cell marked.#

The supplied script repeatGen.pl can extend helical structures to make larger sections of the geometry. Eventually this functionality will be included in the dp_tools package, but for the moment these repeats need to be added to the end of the structure for this older script

2  H
C
1 1    0.2756230044E+01    0.2849950460E+01    0.1794011798E+01
2 1    0.2756230044E+01    0.2849950460E+01    0.3569110265E+00
0 0 0
0.2140932670E+01 18.0 10
3 10

Then repeatGen.pl struct.gen > tmp.gen to make a repeated geometry with the full ring geometry (the usual gen2xyz script can process this: gen2xyz tmp.gen).

This example is can also be represented as a compact supercell, but what happens if the twist angle is not 18 degrees (i.e. a simple rational ratio with 36 degrees)?

Ribbon structures#

[Input: recipes/boundaryconditions/helical/]

Symmetric ribbon geometries can also be represented with helical boundary conditions. Both a conventional supercell and a helical geometry for the same structure are supplied. The helical case has a 180 degree rotation symmetry to make the other side of the ribbon. The supercell has simple translational symmetry along the ribbon with a cell containing twice as many atoms.

Ribbon geometry as a supercell and one possible helical cell consisting of half of the supercell.

The k-points for a helical geometry represent sampling along the screw axis (and a shift):

KPointsAndWeights = HelicalUniform {20 0.5}

In this case generating 20 points along the reciprocal space (and automatically x2 points around the C2 rotation). In this case, the translational operation is directly equivalent to the supercell, so both calculations give equivalent band structures.

Band structure of helical cell, showing band that are symmetric (g) and anti-symmetric (u) under a C2 rotation. In this case, the band structure for a supercell is identical if the labels are ignored.

Figure 61 Band structure of helical cell, showing band that are symmetric (g) and anti-symmetric (u) under a C2 rotation. In this case, the band structure for a supercell is identical if the labels are ignored.#

However, in the case of the helical structure, the k-points for the second C2 symmetry operation allow us to separate bands that are symmetric (g) and anti-symmetric (u) with respect to that operation.

We can of course optimise the atomic coordinates, but also we can twist the structure, by changing the helical angle, to produce geometries that cannot be easily represented as a supercell (a 1 degree twist requires 360 repeats along the axis to be equivalent under translation).

Try twisting the ribbon and relaxing it’s geometry, does the energy go up or down compared to a flat ribbon?

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