Quasiparticle gap versus optical gap

In plain words

Two answers to ‘how much energy does it take to excite this material?’. Adding a free electron and a costs more; creating a bound electron–hole pair with light costs less, and in the difference is large.

Going deeper

Tunnelling measures what it costs to add or remove one electron; light measures what it costs to create a bound electron–hole pair. The difference is the exciton binding energy – 0.55 eV for monolayer MoSe₂ on graphene, compared with about 15 meV in bulk silicon.

Two different questions

The quasiparticle gap is the energy to take an electron out of the material and put another one in, leaving both free to wander: it is what plus inverse photoemission measure, and what a tunnelling tip reads as the separation between the . It is also what calculations mean by the band gap once the self-energy is included.

The optical gap is the energy of the lowest transition that light can drive. Light creates an electron and a hole at the same time and in the same place, so they attract each other and settle into a bound state, the . That bound state lies below the conduction band by the binding energy, so the optical gap is smaller. In bulk silicon the difference is about 15 millielectronvolts and easy to ignore; in a it is not.

Why the difference is so large in 2D

The electric field between an electron and a hole in a thin layer reaches out into the vacuum or the , where there is little to screen it. The pair is also confined to a plane, which pushes them closer together. Both effects strengthen the attraction, and binding energies in the hundreds of millielectronvolts result.

Measuring both gaps on the same sample settles the size. gives the single-particle gap of a monolayer, gives the exciton transition, and the difference for monolayer MoSe₂ on graphene came out at 0.55 eV – orders of magnitude more than in conventional three-dimensional , and in line with –Bethe–Salpeter calculations.

Why the numbers move

Neither gap is a fixed property of the material. Put the same monolayer on a more polarisable substrate, or encapsulate it in hBN, and the surroundings screen both the added charges and the electron–hole attraction. The quasiparticle gap shrinks, the binding energy shrinks with it, and the two changes largely cancel, so the optical gap barely moves. This is why optical spectra look similar across substrates while tunnelling gaps do not.

Adding carriers has a similar effect: a dense electron gas screens the interaction and renormalises the gap downwards. So a quoted gap needs its method attached – tunnelling, photoemission, absorption or emission – along with the substrate, the temperature and the .

For specialists

The quasiparticle gap is the energy to add an electron and a hole independently; the optical gap is smaller by the exciton binding energy.

Where this comes from

Giant bandgap renormalization and excitonic effects in a monolayer transition metal dichalcogenide semiconductor Ugeda et al. · Nature Materials 13, 1091 (2014) cited by 1,870

Electron hole

Usually just called a hole: the empty place an electron leaves behind in an otherwise full band. It behaves like a particle of its own with a positive charge. When a neighbouring electron steps into the gap, the gap moves the other way – like the empty space in a queue of cars, which travels backwards as each car edges forward. Many materials conduct mainly with holes, and chips need both kinds of carrier.

An unoccupied state in an otherwise filled band, usually near the valence-band maximum, treated as a quasiparticle with positive charge and a positive effective mass set by the band curvature. Holes carry the current in p-type semiconductors and give a positive Hall coefficient. In 2D semiconductors good p-type transport is harder to obtain than n-type, largely because metal contacts pin the Fermi level nearer the conduction band.

Full explanation

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.

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Substrate

The base a thin film or flake sits on – often a polished slice of silicon, glass or sapphire. For a crystal one atom thick, the substrate is not just a table: its bumps, stray charges and vibrations reach right into the sheet and change how it conducts and glows. That is why the same material can behave differently on two substrates, and why the flattest, cleanest one – boron nitride – gives the best results.

The material beneath a 2D layer. It acts on the layer through surface roughness, charged impurities, surface optical phonons, dielectric screening, strain and charge transfer, and during growth it sets orientation through epitaxy. Replacing SiO₂ with hexagonal boron nitride, which is atomically flat and nearly free of dangling bonds and charge traps, improved the mobility and charge homogeneity of graphene by close to an order of magnitude.

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Scanning tunnelling microscopy (STM)

A needle sharpened to a single atom is brought so close to a surface that electrons tunnel across the gap – a quantum effect by which they pass through a barrier they could never climb over. The current maps the surface atom by atom, and how it changes with voltage tells you which energies electrons are allowed to have at that exact spot.

Tunnelling between a sharp tip and a conducting surface, giving atomic-resolution topography and, in spectroscopy mode, the local density of states against energy. On 2D materials it resolves defect levels, moiré potentials, edge modes and quasiparticle gaps site by site. The sample has to be clean and conducting, so it is run on epitaxial films or on flakes placed on graphite, in vacuum and often at cryogenic temperature.

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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.

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