Coulomb truncation

In plain words

A correction used in computer simulations of . Simulations usually repeat a sheet endlessly above and below, like the reflections between two facing mirrors, and those copies would wrongly pull and push on each other; truncation cuts that false interaction out.

Going deeper

Plane-wave codes repeat the cell in all three directions, so an isolated sheet is really an infinite stack. Truncating the Coulomb interaction beyond half the cell removes the interaction between those copies exactly, instead of trying to outrun it with vacuum.

The price of periodic boundary conditions

Almost all plane-wave codes assume the simulation cell repeats forever in three dimensions, which is ideal for crystals and awkward for anything that is not one. A is modelled as a slab with vacuum above and below, and that vacuum is never empty enough: the layer polarises its own periodic images, and any long-range field it creates is felt by all of them.

The brute-force remedy is to add more vacuum, but the error falls off only as one over the separation, and every ångström of vacuum costs plane waves. For a charged cell – a charged defect, a slab with net charge – the sum does not converge at all and has to be patched with a compensating background whose contribution then has to be corrected for. The result is calculations that look converged with respect to everything except the one parameter that still matters.

Cutting the tail exactly

The alternative is to change the interaction rather than the geometry. The Coulomb term is multiplied in reciprocal space by a cutoff function that leaves it untouched within the cell and removes it beyond, chosen so that the result is analytic and exact for the periodicity at hand – one shape for a chain, another for a layer, another for a molecule.

The subtlety is that the divergent parts must be handled consistently: the ionic potential and the Hartree term have to be screened in the same way as the electron–electron interaction, and then the divergences cancel. Done properly, the technique is exact rather than approximate, fast, and a small change to an existing code – which is why it has become standard rather than a specialist trick.

Where it changes the answer

It matters most where the long-range Coulomb interaction is the physics. computed with converge painfully slowly with vacuum without it, and the value quoted depends on how much vacuum the author could afford. from the Bethe–Salpeter equation are worse, since a 2D exciton is bound by exactly the interaction the images corrupt. Charged defect formation energies and of polar slabs need it too.

Truncation is not the whole story for a 2D system. The screening of a layer is strongly dependent on the in-plane wavevector and non-analytic as that wavevector goes to zero – the same physics as the Rytova–Keldysh interaction – so a truncated calculation still needs a very dense sampling near the zone centre, or an analytic treatment of the small-wavevector region, before an exciton binding energy is trustworthy. Truncation removes one error; it does not remove that one.

For specialists

A modification of the Coulomb interaction in periodic codes that removes spurious interaction between repeated layers, essential for GW, BSE and charged calculations in 2D.

Where this comes from

Exact Coulomb cutoff technique for supercell calculations Rozzi et al. · Physical Review B 73, 205119 (2006) cited by 498
Truncation of periodic image interactions for confined systems Ismail-Beigi · Physical Review B 73, 233103 (2006) cited by 361

Band structure

The map of the energies an electron may have in a crystal. The allowed energies come in ranges called bands, with forbidden gaps between them – like the decks of a multi-storey car park, where a car can stand on a deck but never between two. Whether a material conducts, gives off light or lets its electrons move easily can all be read from it, and in a layered material the map changes with the number of layers.

The electron energies E(k) of a periodic crystal as functions of crystal momentum, one branch per band, usually plotted along high-symmetry lines of the Brillouin zone. Band edges, slopes and curvatures give the gap, group velocities and effective masses; crossings and inversions give the topology. In layered crystals interlayer hybridisation makes it depend on thickness: the valence-band maximum of MoS₂ moves from Γ in the bulk to K in the monolayer, turning an indirect gap into a direct one.

Full explanation

GW and the Bethe–Salpeter equation

Two calculations that take over where density functional theory, the standard one, stops. The first corrects the energy cost of adding or removing an electron; the second adds the attraction between an electron and the hole it leaves behind, which is what decides the colour of light a thin crystal absorbs.

GW replaces Kohn–Sham eigenvalues with quasiparticle energies from a screened-exchange self-energy, opening the gaps that semi-local DFT underestimates; solving the Bethe–Salpeter equation on top adds the electron–hole interaction and produces optical spectra with bound excitons. In 2D both converge painfully slowly with vacuum size and k-point sampling, because screening is non-analytic near q = 0 – a Coulomb cutoff and dense sampling around the band edge are not optional.

Full explanation