The formula for how strongly an attract each other inside a very thin sheet. Much of the electric field between them passes through the air or material outside the sheet rather than through the sheet itself, so the pull changes with distance in an unusual way – and the pair’s energy levels do not form the neat ladder of a hydrogen atom.
Going deeper
Inside a thin sheet the electric field lines leave the material, so the screening depends on distance. Below the screening length r0 the attraction grows only logarithmically; beyond it, the interaction is the bare Coulomb one.
Screening that runs out
In a bulk the Coulomb interaction between two charges is reduced everywhere by the same factor, the , and that is the whole story. In a sheet one atom thick it is not, because most of the field between two charges does not stay inside the material. Field lines leave into the vacuum or into whatever the sheet is resting on, where nothing screens them.
The result is a screening that depends on separation. Two charges close together – closer than a length r0 set by the sheet’s own polarisability – have their field mostly inside the layer and are screened; far apart, the field is mostly outside and they interact as if the sheet were not there. The interaction that follows is logarithmic at short range and Coulombic at long range, with r0 marking the crossover. An exact analytic form exists, and r0 can be computed from the layer polarisability with standard methods.
A ladder that is not hydrogen’s
The consequence people meet first is that the series in a does not follow the hydrogen pattern. A hydrogenic series has binding energies falling as one over n squared, which is what an exciton in a bulk does. In a monolayer, the tightly bound ground state is small enough to sit inside r0 and feels the screened, logarithmic interaction, while the large excited states extend well beyond r0 and feel the unscreened one. The ladder is therefore stretched relative to hydrogen’s.
This is not a curiosity but the basis of a measurement. Because the excited states converge on the free-particle gap, fitting the observed series with the correct potential is how are extracted from optical spectra – and fitting it with a hydrogenic formula gives the wrong gap and the wrong binding energy. models built on this interaction reproduce full calculations of exciton binding, and extend to the harder three-body problem of .
The environment is one of the parameters
Since much of the field lies outside the sheet, whatever is outside matters. A monolayer on silicon dioxide, suspended, or in boron nitride has three different effective screening lengths and three different exciton binding energies – differences of a hundred millielectronvolts or more, which is not a correction but a redefinition of the numbers.
A partial cancellation saves some of the physics: stronger external screening reduces the exciton binding energy and shrinks the quasiparticle gap by a similar amount, so the optical transition energy moves much less than either. That is why peaks look reassuringly reproducible across while the underlying gap is not, and it is the reason a paper quoting a binding energy has to say what the layer was sitting on. In a periodic calculation the same physics is what exists to get right.
For specialists
The effective electron–hole interaction in a thin sheet: logarithmic at short range, Coulomb-like at long range, which makes 2D exciton series non-hydrogenic.