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Semi-infinite structures

Semi-infinite structures#

[Input: recipes/boundaryconditions/surfaces/Si100]

The same methods used for Electron transport calculations can also be applied to simulate structures like the surface of a semi-infinite crystal, or the ends of polymer chains, nano-wires or nanotubes.

Periodic boundary slab#

A supercell input (Si100_slab.hsd) is provided for the Si (100) surface with buckled dimers on top of a (thin) bulk-like region. In this case there are (obviously?) two surfaces on the top and bottom of the slab, and most of the atoms in between are kept fixed during optimisation.

There are several potential problems with this geometry

  • There are two surfaces, which are potentially close enough to influence each other, either through the bulk of the cell or across the (artificial) vacuum layer separating to the next cell.

  • Charge transfer from the bulk to the surface might be happening in the real system on top of a thick crystal, but this model structure is forced to be charge neutral

  • The dipole moment of the slab can cause problems with self-consistency (this particular example has inversional symmetry, so prevents a dipole occurring).

Green’s function embedding#

The alternative is to represent a semi-infinite bulk structure by replacing the slab with a contact made of bulk material, and simulating atoms on top of this. There are more details in the Electron transport section, but this calculation proceeds in two steps

1. Perform a special periodic calculation to evaluate the bulk
   contact Fermi energy and atomic potentials (see the provided
   'bulk.hsd' input)

2. Use the resulting `shiftcont_bulk.bin` file to calculate the
   surface on top of this semi-infinite bulk (see the provided
   'surface.hsd' input)

There are several unusual features in both calculations, compared to a supercell/periodic input

  • The geometry (see Si_2x1.gen) contains the surface atoms first, then layers of the bulk material (in a specific order).

  • Both calculations have an extra Transport { block, specifying which atoms are bulk and which are surface (contact and device respectively).

Other parts of the bulk calculation input looks like a usual periodic calculation (k-points), but the surface calculation uses some other features:

Solver = GreensFunction{}
Electrostatics = Poisson {
  MinimalGrid [Angstrom] = 0.3 0.3 0.3
  PoissonThickness [AA] = 50
}

This is now using the libNEGF Green’s function solver for an open system, instead of the usual method of finding eigenvalues. Additionally a Poisson solver is used, instead of the usual Ewald method, and in this example it solves the electrostatics in a region extending 50 Angstroms above the top of the surface.

The k-points are also somewhat different, the structure is periodic in the xy plane, so these are like a usual supercell. But in the z direction there should not be any k-point integration, as this is open (a single point with a shift of 0.0 is used):

KPointsAndWeights = SupercellFolding {
  10 0 0
  0 10 0
  0 0 1
  0.5 0.5 0.0
}

(superficially this looks similar to the slab geometry input, but in that case the vacuum ‘slab’ between the silicon surfaces removes the need for k-point integration along z).

The geometry optimisation is also limited to the top layers of the structure (atoms 1 to 16). Unlike the slab calculation, the reason for this is that the analytical forces (evaluated by the Helmann-Feynman theorem) become incorrect close to the contact, but are accurate beyond a few layers of bulk-like material in the device region.

Hence, it’s important to include a buffer of geometrically fixed bulk-like material in the device region below the surface atoms.

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