GW and the Bethe–Salpeter equation

In plain words

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

Going deeper

Semi-local DFT underestimates the gap, GW corrects it to the energy of adding or removing an electron, and the Bethe–Salpeter equation adds the electron–hole attraction that sets the optical gap. For a 2D sheet, periodic copies screen each other unless the Coulomb interaction is truncated, and results then converge only slowly with vacuum size.

Why DFT gaps are too small, and what GW fixes

The band energies of density functional theory are not, strictly, the energies needed to add or remove an electron, and the common semi-local functionals underestimate substantially, often by a third or more. Hybrid functionals help, but include screening only in an averaged way.

GW, introduced by Hedin in 1965, calculates energies directly. The self-energy – the correction an electron feels from the response of all the others – is approximated as the product of the one-particle Green’s function G and the interaction W. Usually GW is applied once on top of a DFT calculation, called G₀W₀, or iterated partly to self-consistency. The result is the , the quantity measured by and .

Adding the exciton

Light does not measure the quasiparticle gap. It creates an electron and a hole that attract each other, and the Bethe–Salpeter equation, solved with GW energies and the screened interaction, finds those correlated pairs: energies, binding energies and a full absorption spectrum.

In this step is not a small correction. Free-standing MoS₂ is calculated to have excitons bound by close to 1 eV; a or screens the attraction and reduces it. A historical warning follows: semi-local DFT gaps of monolayers often land near the measured optical gap, but only because two large errors, the missing self-energy and the missing exciton binding, roughly cancel.

Why 2D calculations are expensive

Plane-wave codes treat a sheet as a stack of periodic copies separated by vacuum. The long-range Coulomb interaction couples those copies, so each layer is screened by its images, and gaps and binding energies creep towards their true values only slowly as the vacuum grows. Truncating the Coulomb interaction between copies removes the problem and should be standard.

Screening in 2D also changes rapidly at small wavevectors, so the Brillouin zone has to be sampled densely near its centre, and convergence with the number of empty bands and the cutoff is slow. GW and BSE cost far more than DFT, and is essential for TMDCs. Results only mean something with the starting functional, the level of self-consistency, the truncation, the k-point grid and the treatment of the substrate stated.

For specialists

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.

Where this comes from

New method for calculating the one-particle Green’s function with application to the electron-gas problem Hedin · Physical Review 139, A796 (1965) cited by 5,512
Electron-hole excitations and optical spectra from first principles Rohlfing and Louie · Physical Review B 62, 4927 (2000) cited by 1,955
Optical spectrum of MoS₂: many-body effects and diversity of exciton states Qiu et al. · Physical Review Letters 111, 216805 (2013) cited by 1,616

Density functional theory (DFT)

The workhorse calculation of materials physics. Rather than following every electron, it works with the electron density and a recipe for how electrons avoid one another – which makes a whole crystal cheap enough to compute, and makes the accuracy depend entirely on that recipe.

Ground-state electronic structure from the density, exact in principle by Hohenberg–Kohn and made practical by the Kohn–Sham equations with an approximate exchange–correlation functional. Two dimensions need care: a vacuum gap wide enough that periodic images stop interacting or a truncated Coulomb interaction, a dispersion correction for interlayer binding, and the knowledge that semi-local functionals underestimate gaps – hybrid functionals narrow that error at far higher cost, and quantitative gaps and optical spectra need GW and Bethe–Salpeter on top.

Full explanation

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

Quasiparticle

An electron moving through a crystal is never alone: it pushes and pulls on all the other electrons and atoms around it, and drags a little cloud of disturbance along. Physicists treat the electron together with its cloud as a new particle – a quasiparticle – with its own mass and lifetime. The idea goes further: holes, excitons, vibrations of the lattice and waves of spins also behave like particles, and describing a solid as a gas of such quasiparticles is how most of its properties are understood.

An elementary excitation of an interacting many-body system that behaves like a particle with a defined energy–momentum relation, charge or spin, and a finite lifetime, as in Landau’s Fermi-liquid theory. Dressed electrons and holes (renormalised by screening, phonons or correlations, with effective masses that can reach hundreds of free-electron masses in heavy-fermion systems), excitons, trions and polarons, and bosonic collective modes such as phonons, magnons, plasmons and polaritons are all quasiparticles. Their energies are poles of the Green’s function; the GW approximation computes quasiparticle band structures, which in 2D semiconductors are strongly renormalised by the reduced screening, so that the quasiparticle gap exceeds the optical gap by the large exciton binding energy.

Full explanation

Dielectric screening

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 2D materials, and why whatever a sheet rests on changes its properties.

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 dielectric 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.

Full explanation

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.

Full explanation

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.

Full explanation

Dielectric

An electrical insulator used for what it does in an electric field rather than for carrying current – the thin layer inside a capacitor, or the insulator that separates a transistor’s gate from its channel. A good dielectric blocks current completely, withstands a strong field without breaking down and, for gates, has a high permittivity, so that a voltage on the gate pulls many charges into the channel. For 2D transistors the dielectric is as hard to get right as the 2D layer itself.

An insulator characterised by its permittivity κ, band gap and band offsets to the channel, breakdown field, leakage, and density of fixed charge and traps at the interface and in the bulk (border traps). Silicon technology moved from SiO₂ (κ ≈ 3.9) to hafnium-based high-κ oxides (κ ≈ 20–25) to keep gate control at sub-nanometre equivalent oxide thickness. For 2D channels, hBN gives clean, trap-poor interfaces but a low κ of about 3–4 and is hard to grow at wafer scale; ALD oxides nucleate poorly on basal planes; alternatives include crystalline CaF₂, native oxides such as Bi₂SeO₅ on Bi₂O₂Se, and molecular crystals. Reliability – hysteresis, bias-temperature instability and time-dependent breakdown – is the yardstick for any dielectric meant for products.

Full explanation