The way a material weakens the pull between two charges inside it: its own electrons shift slightly and partly cancel the field between them. A sheet one atom thick has almost nothing around it to do this, so charges in it attract each other far more strongly than in a thick crystal – which is why light makes tightly bound pairs in , and why whatever a sheet rests on changes its properties.
Going deeper
Left: inside a thick crystal the atoms around an electron and a hole shift their charge and cancel much of the field between them, so the pair is weakly bound. Right: in a single layer most of the field runs through the space above and below the sheet, where nothing cancels it, so the pair stays strongly bound – and whatever surrounds the sheet takes part in the screening.
What screening does
Put a charge inside a material and the material responds. In an each atom’s electron cloud shifts a little towards or away from it, so every atom becomes a small dipole pointing against the field; in a metal the free electrons crowd round a positive charge and draw away from a negative one. Either way the field further away is weakened. In an insulator the factor is the dielectric constant – about 12 for silicon and 4 for silicon dioxide. In a good metal the field is cancelled almost completely within an ångström or so, and in a lightly doped within a length that grows as the falls.
Screening is why an it leaves behind barely bind in an ordinary semiconductor. Their attraction is the one that holds a hydrogen atom together, but in silicon it is divided by the dielectric constant and weakened again by the carriers’ small , so a binding of 13.6 eV in hydrogen becomes about 15 meV – less than the 26 meV of room-temperature thermal energy, and the pair falls apart as soon as it forms.
Why a thin sheet screens poorly
In a sheet a fraction of a nanometre thick there is little material to polarise. The field between two charges a few nanometres apart runs mostly outside the sheet, through the vacuum or the , so the sheet screens charges that are close together but hardly those far apart: the effective dielectric constant depends on the distance, as the describes. Electron–hole pairs in a are therefore small and strongly bound. In monolayer WS2, Chernikov and colleagues measured a binding energy of about 0.32 eV from the series of excited states, whose spacing departs from hydrogen’s because of exactly this distance dependence.
The same weakness makes the larger: an electron added to a poorly screened sheet costs more energy, so the of an isolated monolayer is well above what a calculation tuned to bulk screening would give. Electrons also feel one another more strongly, which is part of why – Wigner crystals, correlated insulators in bilayers – are easier to reach in two dimensions.
The surroundings as a design parameter
Because much of the field lies outside, what surrounds the sheet is part of the material. Moving a monolayer from vacuum onto silicon dioxide, into boron nitride or onto graphene lowers both its quasiparticle gap and its exciton binding energy, by similar amounts, so the optical transition moves far less than either. Raja and colleagues shifted the band gap of WS2 and WSe2 monolayers by hundreds of meV this way, changing only what lay around them, and used it to give a single, chemically uniform layer a band gap that changes from one region to the next.
Two practical consequences follow. A binding energy or a quasiparticle gap means little without the substrate and it was measured on, and a calculation has to include that environment rather than treat the layer as free-standing. And free carriers screen too: doping a monolayer, electrically or by , weakens the exciton binding and shrinks the gap, and at a high enough density the excitons dissolve into a plasma of free electrons and holes.
For specialists
The reduction of the Coulomb interaction by the polarisation of bound electrons and ions and, in metals and doped layers, by free carriers, described by a function ε(q, ω). In a monolayer the field lines between distant charges run through the surroundings, so ε(q) approaches 1 as q → 0 and the screening is non-local and set by the environment. Exciton binding energies reach hundreds of meV, and a change of substrate or encapsulation shifts the quasiparticle gap by 100 meV or more while the optical gap barely moves.