Theory track · for people who compute

Theory & computation

For people who compute and model: DFT and beyond-DFT practitioners, many-body theoreticians, and anyone building tight-binding or continuum models of van der Waals materials. This track follows the questions where theory is ahead of, behind, or in open disagreement with experiment – and it names the methodological traps that make 2D calculations harder than they look.

12 open questions12 methods with traps25 codes and databases13 papers

This page is written for specialists. For a plain-language introduction, start with the basics.

You are reading as an experimentalist. This is the theory track; the experiment track is the one written for you.

You are reading as an engineer. This is the theory track; the engineering track is the one written for you.

A moiré pattern computed from two identical lattices turned 6° against each other. Where the dots of both layers coincide (circled) the stacking is AA; in between they interleave. The large pattern repeats with period λ = a / (2 sin θ/2), about 9.6 lattice constants at this angle – and about 52 at graphene’s magic angle of 1.1°. Moiré superlattice in the glossary

Open questions

12 places where theory is ahead of experiment, behind it, or in open disagreement with it.

Moiré flat bands and correlated phases

What is happening

Twisted and aligned produce whose bandwidth can be tuned below the Coulomb energy.

The width of a band measures how easily electrons move through the crystal. When that width falls below the energy it costs two electrons to sit near each other, motion stops mattering and interactions take over – which is what happens in twisted bilayer graphene near 1.1°. Flat band in the glossary

Read moreShow less

, multilayers and heterobilayers such as WSe₂/WS₂ now show , , orbital magnetism, generalised Wigner crystals and – all with filling and displacement field. Twisted bilayer WSe₂ superconducts too, the first clear moiré superconductor outside graphene.

Why it is hard

The relevant energy scales are meV, small compared with typical errors. Supercells contain thousands to tens of thousands of atoms, lattice relaxation and heterostrain reshape the bands, and competing symmetry-broken orders are separated by fractions of a meV. Continuum models need parameters – interlayer tunnelling, the AA/AB tunnelling ratio, – that are themselves uncertain.

What would settle it

Parameter-free predictions of which order wins at a given filling and twist angle, tested against spectroscopy (, , compressibility) and not only transport – and an agreed superconducting mechanism that predicts how Tc responds to screening and twist-angle before it is measured.

Fractional Chern insulators at zero magnetic field

What is happening

In 2023–2024 fractionally quantised states – fractional Hall resistance with no applied magnetic field – were observed in twisted bilayer MoTe₂ and in rhombohedral pentalayer graphene aligned with hBN.

A fractional Chern insulator is the lattice version of a fractional quantum Hall state: the same fractionally charged excitations, but produced by the band structure of a twisted stack rather than by a laboratory magnet. Fractional Chern insulator in the glossary

Read moreShow less

They are lattice analogues of fractional states, stabilised by flat Chern bands with favourable quantum geometry.

Why it is hard

Whether a band favours fractionalisation depends on the uniformity of its and , not just its flatness. Exact diagonalisation and DMRG are restricted to small systems. In pentalayer graphene the mechanism itself is debated: the moiré potential may be too weak to matter, pointing to an interaction-driven Chern band. Predicted non-Abelian states at even-denominator fillings remain unconfirmed.

What would settle it

Direct evidence for fractional charge or anyonic statistics – interferometry, shot noise or thermodynamic probes – together with a predictive theory of which stacking, angle and displacement field produce which fractional states.

Excitons and quasiparticle gaps in reduced dimensions

What is happening

Weak, non-local screening in produces of hundreds of meV, non-hydrogenic Rydberg series and strong environmental renormalisation of both and exciton energies.

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. Quasiparticle gap versus optical gap in the glossary

Read moreShow less

in heterobilayers, dark and momentum-indirect excitons, and exciton– in doped samples are where much of the current work is.

Why it is hard

converges slowly in 2D with k-points, empty bands and vacuum size, and the screening changes abruptly as q → 0, which calls for and special small-q integration. and shift gaps by hundreds of meV, so a free-standing number is not what is measured. Exciton– coupling sets linewidths and sidebands but is expensive to compute from first principles.

What would settle it

Systematic benchmarks of GW–BSE against scanning tunnelling spectroscopy gaps and optical spectra measured on the same well-characterised encapsulated samples, including temperature-dependent linewidths from first-principles exciton–phonon theory.

Magnetic order in two dimensions

What is happening

Intrinsic magnets – CrI₃, near-Heisenberg Cr₂Ge₂Te₆, XY-like CrCl₃ and NiPS₃, itinerant Fe₃GeTe₂ – let theory test finite-temperature order in 2D directly.

The theorem is about how cheap long-wavelength spin waves are. With a continuous symmetry there is no minimum energy to excite them, and in two dimensions there are enough such modes at any finite temperature to destroy long-range order. Anisotropy opens a gap, and the argument no longer applies. Mermin–Wagner theorem in the glossary

Read moreShow less

Stacking-dependent , , magnetic proximity effects in heterostructures and predicted two-dimensional altermagnets have become quantitative questions.

Why it is hard

Long-range order in 2D exists only because of , so small –orbit-derived terms – single-ion anisotropy, anisotropic and Kitaev-type exchange – decide the outcome. DFT+U exchange constants vary strongly with U and with the double-counting choice, and interlayer exchange sits at the meV level, comparable to dispersion-correction errors. require classical or quantum rather than mean-field estimates.

What would settle it

Spin Hamiltonians that simultaneously reproduce measured magnon spectra ( or scattering), thickness-dependent ordering temperatures and stacking-dependent for the same material.

Sliding and in-plane ferroelectricity

What is happening

Layers that are individually non-polar become when stacked with a particular registry: parallel-stacked bilayer hBN, rhombohedral TMDC bilayers and few-layer WTe₂ switch their polarisation by interlayer sliding rather than ionic displacement.

Sliding ferroelectricity stores its state in the registry between layers rather than in the position of an atom within a unit cell. Sliding one layer by a single lattice site reverses the polarisation, and no bond has to break. Sliding ferroelectricity in the glossary

Read moreShow less

Separately, the group-IV monochalcogenides and In₂Se₃ are ferroelectric within each layer; in SnTe one thick, switching was shown in 2016.

Why it is hard

Sliding polarisation is orders of magnitude smaller than in perovskite ferroelectrics and arises from subtle , which is sensitive to both the dispersion correction and the exchange–correlation functional. Switching proceeds through domain walls with soliton-like structure that need large supercells. In a metal such as WTe₂, even defining a switchable polarisation requires care.

What would settle it

First-principles switching pathways and domain-wall energies that reproduce measured coercive fields and retention, and a clear account of how sliding polarisation couples to excitons, superconductivity or magnetism within the same stack.

Topological phases in 2D crystals

What is happening

Monolayer 1T′-WTe₂ remains the cleanest natural , bismuthene on SiC shows a large topological gap, and heavy , layers and strained monolayers are proposed as further candidates.

In a two-dimensional topological insulator, the quantum spin Hall state, the interior is insulating while each edge carries two counter-propagating channels of opposite spin. Their bands cross inside the bulk gap, and a non-magnetic defect cannot scatter a carrier backwards without flipping its spin. Topological insulator in the glossary

Read moreShow less

Higher-order topological phases with corner states and the topology of moiré bands are newer questions.

Why it is hard

in small-gap systems depend on the exchange–correlation treatment: PBE can predict a topological where hybrid functionals or GW give a trivial insulator, or the reverse. Substrates hybridise with , and interactions can turn an inverted-band insulator into an , as argued for WTe₂. Symmetry indicators and Wilson loops must be evaluated for the real, supported geometry.

What would settle it

Transport showing quantised edge conductance over longer channels and at higher temperatures, and calculations that predict gap sizes within the spread of STS measurements once substrate and correlation effects are included.

Superconductivity in the 2D limit

What is happening

Monolayer NbSe₂ and gated MoS₂ are Ising superconductors whose in-plane critical fields exceed the Pauli limit.

In a monolayer that lacks inversion symmetry, spin–orbit coupling acts like a magnetic field pointing out of the sheet, with opposite sign in opposite valleys. Cooper pairs built from those states are hard for an in-plane field to break. Ising superconductivity in the glossary

Read moreShow less

WTe₂ superconducts under gating alongside its topological edges, and twisted and rhombohedral graphene superconduct at very low , in some cases next to spin- and -polarised metals. Twisted bilayer WSe₂ also superconducts, at twist angles of 3.65° and 5°.

Why it is hard

Eliashberg theory with DFPT phonons works for NbSe₂-like metals but depends on anharmonicity and softening. In systems, strong Coulomb repulsion, quantum geometry and proximity to symmetry-broken phases undermine the assumptions of conventional Migdal–Eliashberg theory. physics, disorder and inhomogeneity complicate any comparison with measured Tc.

What would settle it

Pairing-symmetry measurements – tunnelling spectroscopy, phase-sensitive and Josephson interferometry – combined with calculations that predict isotope, screening and twist-angle dependences in advance.

Defects, dopants and quantum emitters

What is happening

decide device performance – , substitutional oxygen, antisites – and also create useful quantum systems: room-temperature and optically addressable spin defects in hBN, and -localised emitters in WSe₂.

A quantum emitter releases photons one at a time. In a monolayer, excitons roll into a strain well – for instance where the layer is draped over a pillar – and emit from that one spot; the proof is a dip in coincidences at zero delay, well below one half. Quantum emitter in the glossary

Read moreShow less

Why it is hard

Charged-defect energetics in 2D need dedicated electrostatic corrections, because standard 3D image-charge schemes fail in slab geometry, and charge transition levels depend strongly on the functional. Identifying an emitter means matching zero-phonon line, , polarisation and hyperfine structure simultaneously – and many candidate structures give similar numbers.

What would settle it

Atomically resolved identification – STM or combined with optical or spin spectroscopy on the very same defect – matched against first-principles fingerprints that include vibronic coupling. For the visible-range emitters in hBN, a definitive structural assignment; only the boron-vacancy spin centre is settled.

Electron–phonon coupling and intrinsic mobility

What is happening

First-principles Boltzmann transport with Wannier-interpolated matrix elements predicts intrinsic limits of 2D and is used to screen candidate channels.

Scattering mechanisms add up: the inverse mobilities from impurities and from phonons sum, so the lower limit wins at each temperature. How the device is wired decides whether the contacts are counted as part of the material – in a two-terminal measurement they are, in a Hall bar they are not. Charge carrier mobility in the glossary

Read moreShow less

Measured mobilities usually fall far below these limits.

Why it is hard

Long-range Fröhlich and couplings in 2D need a dedicated treatment of the q → 0 limit with 2D Coulomb truncation; getting it wrong changes predicted mobilities several-fold. Calculations describe ideal crystals, whereas devices are limited by remote phonons from , charged impurities, defects and contacts.

What would settle it

Mobility predictions that include substrate phonons, dielectric screening and realistic defect densities, and still match the best encapsulated devices – plus a community benchmark for 2D mobility calculations.

Contacts, band alignment and interfaces

What is happening

has long limited 2D transistors. Semimetal contacts brought it down to 123 Ω·µm with bismuth on MoS₂ (2021) and 42 Ω·µm with antimony (2023), close to the quantum limit, but only for n-type contacts; contacts are still far worse.

In an ideal contact the barrier height would follow the metal’s work function. In practice, interface states fix the Fermi level near one position, so swapping the metal changes the barrier very little – the measured slope for monolayer MoS₂ is about 0.1 instead of 1. Fermi-level pinning in the glossary

Read moreShow less

Theory is used to design van der Waals contacts, semimetal contacts (Bi, Sb) that suppress , and charge-transfer doping layers, and to predict in heterostructures.

Why it is hard

combines metal-induced gap states, interface chemistry, defects and dipoles. Periodic interface models need supercells that are either impractically large or artificially strained, and semilocal misplace Schottky barriers unless corrected with hybrids or GW.

What would settle it

Predicted Schottky barrier heights and contact resistances that match statistically robust device data across several metals and semiconductors, including temperature-dependent transport.

Machine-learned potentials and large-scale simulation

What is happening

Machine-learned interatomic potentials (equivariant message-passing models, Gaussian-process and deep-potential approaches) make near-DFT-quality feasible for moiré relaxation, thermal transport, growth and defect dynamics.

Read moreShow less

Universal potentials trained on large databases give usable results without training your own, but they systematically soften the energy surface – phonon frequencies and barriers come out too low – and inherit the errors of the PBE data they were trained on.

Why it is hard

Potentials extrapolate poorly outside their training data – and unusual stacking registries, bond breaking during growth, charged defects and long-range dispersion are exactly where 2D problems live. Interlayer interactions are weak and must be learned to sub-meV-per-atom accuracy to get stacking energies and sliding barriers right.

What would settle it

Open benchmarks for 2D-specific properties – stacking energies, relaxation in twisted bilayers, phonon dispersions including flexural modes – and uncertainty-aware potentials that flag extrapolation during a simulation.

High-throughput discovery and 2D databases

What is happening

Computational screens and 2D databases catalogue thousands of candidate monolayers with stability, , magnetism and topology, and increasingly feed generative models and active-learning discovery loops.

Read moreShow less

Why it is hard

Databases rely on semilocal functionals and idealised structures; stability criteria differ between databases; and exfoliability is computed from binding energies that depend on the dispersion correction. Most predicted materials have no synthesis route, and synthesisability is not captured by formation energy alone.

What would settle it

Synthesisability metrics validated against experiment, beyond-DFT data (GW, hybrids) at scale, and closed-loop studies in which computationally proposed are actually made and measured.

Methods and traps

12 methods that two-dimensional systems break in ways bulk systems do not, with what to do instead.

Band gaps: semilocal DFT, hybrids and GW

Used forElectronic structure, band alignments, defect levels

The trapPBE underestimates gaps substantially. Hybrid functionals tuned for bulk screening do not capture the weaker screening of an isolated layer, and the measurable gap depends on the dielectric environment – so a free-standing GW gap is not directly comparable to a measurement on a substrate.

What to doUse PBE for geometries and trends, hybrids for alignments, and GW with Coulomb truncation for quantitative gaps. State the dielectric environment explicitly and, for encapsulated or supported layers, include substrate screening – for example through a quasi-2D dielectric model.

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. GW and the Bethe–Salpeter equation in the glossary

MoS₂: one material, three band gaps

Bulk crystal: indirect gap1.2–1.3 eV
Monolayer: optical gap (A exciton), what light measures1.85–1.9 eV
Monolayer: quasiparticle gap, what tunnelling measures and GW computes2.2–2.5 eV

The two monolayer gaps differ by the exciton binding energy: about 0.3–0.65 eV for these values, which depend on the substrate beneath the layer. A free-standing layer is screened less and binds its excitons more strongly, close to 1 eV as in the drawing above. Semilocal DFT (PBE) falls well short of the quasiparticle gap. The numbers are those on the MoS₂ page.

Van der Waals corrections

Used for, binding and exfoliation energies, stacking and sliding energetics

The trapD2, D3(BJ), D4, Tkatchenko–Scheffler, many-body dispersion and nonlocal vdW-DF/rVV10 functionals give interlayer binding energies differing by tens of percent and spacings by several percent – enough to change band gaps, interlayer exchange or preferred stacking in sensitive systems. Pairwise schemes miss many-body screening in layered metals.

What to doBenchmark against RPA results or experimental lattice constants for your material class, report the scheme used, and check that the conclusion – stacking preference, magnetic ground state, gap character – survives a change of scheme.

Van der Waals attraction comes from electrons in one layer fluctuating in step with those in the next. It is a hundred times weaker than the bonds inside a layer, which is why layered crystals can be split apart and restacked at will. Van der Waals force in the glossary

Slab geometry, vacuum and dipole corrections

Used forAny periodic calculation of a monolayer or heterostructure

The trapVacuum that is adequate for total energies can be badly insufficient for GW, BSE and charged calculations. Asymmetric slabs – Janus layers, adsorbates, one-sided heterostructures – create an artificial field across the vacuum unless a dipole correction is applied.

What to doConverge vacuum separately for each property, use Coulomb truncation for excited-state and charged calculations, and apply a dipole correction to any slab lacking perpendicular to the plane.

Coulomb truncation and the q → 0 limit

Used forGW, BSE, polar phonons and electron–phonon coupling in 2D

The trapIn 2D the dielectric function approaches 1 as q → 0 and the screened interaction varies rapidly at small q. Standard 3D small-q treatments and coarse q-grids then over- or under-screen, and results drift with vacuum size when no truncation is used.

What to doCombine 2D Coulomb truncation with analytical or subsampled small-q integration, and check explicitly how quasiparticle gaps and exciton binding energies converge with q-grid density.

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. Coulomb truncation in the glossary

k-point sampling for Dirac and valley physics

Used forMetals and semimetals, Berry-curvature properties, TMDC optical spectra

The trapLinear or strongly curved bands near K and K′ and small demand dense sampling; coarse grids give wrong densities of states, conductivities and exciton binding energies, and Berry-curvature integrals converge slowly near band crossings.

What to doConverge the quantity you report, not the total energy. Use Wannier interpolation for dense grids and non-uniform or patch-refined sampling around the valleys for BSE.

Spin–orbit coupling and non-collinear magnetism

Used forHeavy-element TMDCs, topological materials, magnetic anisotropy

The trapScalar-relativistic calculations miss valence-band valley splittings of ~150–450 meV across the MoS₂–WSe₂ family, band inversions and magnetic anisotropy. Anisotropy energies range from tens of µeV to about a millielectronvolt per magnetic atom and are easily lost in numerical noise.

What to doInclude self-consistently for W-, Bi-, Sb-, Pb- and Te-containing systems and for any anisotropy or topology question. Converge anisotropy energies with dense k-grids and tight thresholds, comparing magnetisation directions explicitly.

Left: in the electron’s own frame the charged nucleus circles it, forming a current loop whose magnetic field acts on the electron’s spin – the origin of spin–orbit coupling. Right: in a TMDC monolayer this splits the top of the valence band at the K point into two bands with opposite spin, by about 150 meV in MoS₂ and 450 meV in the heavier WSe₂. Spin–orbit coupling in the glossary

DFT+U for correlated d electrons

Used for, Cr-, V- and Mn-based , oxides and defects

The trapExchange constants, magnetic ground states and even metallic versus insulating character can change with U by more than the effect under study, and U values borrowed from bulk oxides are rarely justified.

What to doDetermine U from linear response or constrained RPA where possible, present results as a function of U, and validate against measured moments, gaps or magnon spectra. Consider DMFT for strongly correlated systems.

Phonons, imaginary modes and flexural physics

Used forDynamical stability, thermal transport, Raman assignment, electron–phonon coupling

The trapSmall imaginary frequencies near Γ in the flexural (ZA) branch often signal insufficient sampling, supercell size or violated rotational invariance rather than true instability; conversely, electronic smearing can hide a genuine charge-density-wave instability. Harmonic phonons fail for strongly anharmonic materials such as SnSe and perovskites.

What to doEnforce acoustic sum and rotational invariance rules, test supercell size and smearing, and use temperature-dependent effective potentials or stochastic self-consistent harmonic approximations for anharmonic systems.

The phonon branches of a sheet. In-plane acoustic waves (LA, TA) rise linearly with wavevector; the flexural ZA wave, in which the sheet ripples out of plane, rises with its square, because bending a thin sheet costs little energy. These soft flexural modes are a signature of two-dimensional crystals. Phonon in the glossary

Moiré supercells versus continuum and tight-binding models

Used forTwisted bilayers, lattice-mismatched heterostructures

The trap supercells at small angles contain thousands of atoms and impose artificial strain to achieve periodicity. Continuum models are cheap but rely on fitted parameters – tunnelling amplitudes, relaxation, displacement-field response – that may not transfer between angles or stackings.

What to doRelax structures with accurate force fields or machine-learned potentials, extract continuum or Wannier parameters from DFT on the relaxed geometry, and cross-check key conclusions with large-scale tight binding before adding interactions.

Twisted bilayer graphene repeats exactly only at certain angles, where cos θ = (3m² + 3m + ½)/(3m² + 3m + 1), and the smallest repeating cell then holds 4(3m² + 3m + 1) atoms (Lopes dos Santos, Peres and Castro Neto, 2007). At the magic angle that is 11,908 atoms, and it keeps growing as the angle shrinks. A continuum model costs the same at every angle; its price is the fitted parameters named above.

Many-body methods for flat bands

Used forCorrelated insulators, magnetism, superconductivity and fractional states in moiré systems

The trapHartree–Fock favours and neglects fluctuations; exact diagonalisation is limited to small clusters and can miss competing crystalline orders. Results depend on the band projection, on how the interaction is subtracted to avoid double counting, and on the assumed screening.

What to doCompare several methods – Hartree–Fock, DMRG, exact diagonalisation, sign-free quantum Monte Carlo – on the same projected model, state the subtraction scheme, and test sensitivity to the dielectric constant and the number of bands retained.

Quantum transport with non-equilibrium Green’s functions

Used forContacts, , transistor simulation

The trapResults are dominated by the contact and electrostatic model. Hamiltonians taken from semilocal DFT inherit gap errors that shift barriers, and ballistic simulations ignore the phonon scattering that sets room-temperature performance.

What to doUse Wannier or tight-binding Hamiltonians with corrected gaps, solve electrostatics self-consistently with realistic dielectrics, and add scattering self-energies before comparing with measured on-currents.

Point-defect calculations in 2D

Used forFormation energies, charge transition levels, emitters and dopants

The trapWith the uniform compensating charge of a 3D-periodic code, the formation energy of a charged defect in a slab grows without limit as the vacuum widens, and bulk corrections do not remove this. Chemical-potential choices change formation-energy rankings, and semilocal functionals misplace levels and over-delocalise defect states.

What to doUse 2D-specific charge corrections or extrapolation schemes, report chemical-potential ranges, use hybrid functionals for defect levels, and compute zero-phonon lines and Huang–Rhys factors when comparing with optics.

Codes and databases

25 tools and data sources, with what each is for and how it is licensed.

Latest releases · checked 5 Oct 2026

Each code's newest published release, read from its public release page once a day. Not every code publishes releases there, and some release without an announcement.

Codes · 16

VASP

Plane-wave PAW DFT with hybrids, GW, BSE and many dispersion corrections

Commercial licence

Quantum ESPRESSO

Plane-wave DFT and DFPT phonons, with a built-in 2D Coulomb cutoff

Open source (GPL)

ABINIT

Plane-wave DFT, DFPT, GW and electron–phonon capabilities

Open source (GPL)

FHI-aims

All-electron DFT with numeric atomic orbitals; hybrids and many-body dispersion at scale

Licence required

CP2K

Gaussian and plane-wave DFT for large systems and ab initio molecular dynamics

Open source (GPL)

GPAW

Python PAW code with GW, BSE and tools used to build the C2DB database

Open source (GPL)

Yambo

Many-body perturbation theory: GW, BSE and real-time excited-state dynamics

Open source (GPL)

BerkeleyGW

GW and GW–BSE, with Coulomb truncation and subsampling schemes suited to 2D

Open source

Wannier90

Maximally localised Wannier functions for tight-binding models and interpolation

Open source (GPL)

WannierTools

Topological invariants, surface and edge states from Wannier tight-binding models

Open source

Z2Pack

Topological invariants via hybrid Wannier charge centres

Open source

phonopy

Phonons by finite displacements; phono3py adds lattice thermal conductivity

Open source (BSD)

EPW

Electron–phonon coupling, transport and superconductivity via Wannier interpolation

Open source (GPL)

Kwant

Quantum transport in tight-binding systems

Open source (BSD)

LAMMPS

Classical and machine-learned molecular dynamics, including registry-dependent interlayer potentials

Open source (GPL)

MACE

Equivariant machine-learned interatomic potentials, including foundation models

Open source (MIT)

Workflow and automation · 2

pymatgen

Python materials analysis library underpinning the Materials Project

Open source (MIT)

AiiDA

Workflow engine with full provenance tracking for high-throughput calculations

Open source (MIT)

Databases · 7

C2DB

Computational 2D Materials Database: stability, electronic, magnetic, optical and topological properties of thousands of monolayers

Open access

Materials Cloud MC2D

Exfoliable 2D materials identified by computational screening of known 3D compounds

Open access

2DMatPedia

2D materials generated by top-down exfoliation and bottom-up substitution screening

Open access

JARVIS-DFT

NIST database including many monolayers with optB88vdW and meta-GGA properties

Open access

Materials Project

Bulk parent compounds, formation energies and phase stability

Open access (free account)

NOMAD

Repository and archive of raw calculation data with FAIR metadata

Open access

Topological Materials Database

Topological classification of known crystals by topological quantum chemistry

Open access

Searches to follow

20 standing arXiv searches for this track. They are part of what fills this site’s news feed, and each link opens the live feed from the arXiv API for a feed reader to subscribe to; the query itself is written out so you can change it.

2D magnetism

Spin models, anisotropy and magnons in van der Waals magnets

(cat:cond-mat.mtrl-sci OR cat:cond-mat.str-el) AND (abs:"two-dimensional magnet" OR abs:"van der Waals magnet" OR abs:"CrI3" OR abs:"Fe3GeTe2" OR abs:"CrSBr")

Open the live feed

2D superconductivity theory

Pairing in Ising, gated and moiré superconductors

cat:cond-mat.supr-con AND (abs:"two-dimensional" OR abs:"monolayer" OR abs:"twisted" OR abs:"Ising superconductivity")

Open the live feed

Machine learning for 2D materials

Learned potentials and data-driven discovery for layered materials

cat:cond-mat.mtrl-sci AND (abs:"machine learning" OR abs:"neural network" OR abs:"interatomic potential" OR abs:"high-throughput") AND (abs:"two-dimensional" OR abs:"van der Waals")

Open the live feed

Moiré correlated phases

Correlated-electron theory and experiment in moiré systems

cat:cond-mat.str-el AND (abs:moire OR abs:"twisted bilayer")

Open the live feed

Fractional Chern insulators

Zero-field fractional states in twisted MoTe₂ and rhombohedral graphene

abs:"fractional Chern" OR abs:"fractional quantum anomalous Hall"

Open the live feed

GW–BSE and excitons in 2D

Many-body excited-state calculations for layered materials

(cat:cond-mat.mtrl-sci OR cat:cond-mat.mes-hall) AND (abs:"Bethe-Salpeter" OR abs:"GW approximation") AND (abs:monolayer OR abs:"two-dimensional")

Open the live feed

Sliding and 2D ferroelectricity

Stacking-engineered and intrinsic ferroelectric order in thin layers

abs:"sliding ferroelectricity" OR abs:"sliding ferroelectric" OR abs:"two-dimensional ferroelectric"

Open the live feed

Topology in 2D

Quantum spin Hall and higher-order topological phases in atomically thin crystals

(cat:cond-mat.mes-hall OR cat:cond-mat.mtrl-sci) AND (abs:"quantum spin Hall" OR abs:"higher-order topological") AND (abs:monolayer OR abs:"two-dimensional")

Open the live feed

Defects and emitters from first principles

Defect energetics and quantum-emitter identification

(abs:"hexagonal boron nitride" OR abs:"transition metal dichalcogenide") AND abs:defect AND (abs:"first-principles" OR abs:"density functional")

Open the live feed

Electron–phonon and mobility

Intrinsic transport and phonon-limited mobility of 2D semiconductors

cat:cond-mat.mtrl-sci AND abs:"electron-phonon" AND (abs:monolayer OR abs:"two-dimensional")

Open the live feed

Kitaev magnets

α-RuCl₃ and other candidates for the bond-dependent exchange of the Kitaev honeycomb model

abs:RuCl3 OR abs:RuBr3 OR abs:"Kitaev magnet" OR abs:"Kitaev material" OR abs:"Kitaev spin liquid"

Open the live feed

Magnetic topological insulators

MnBi₂Te₄ and its relatives, where magnetism and band topology meet in one crystal

abs:MnBi2Te4 OR abs:MnSb2Te4 OR abs:MnBi4Te7 OR abs:"intrinsic magnetic topological insulator"

Open the live feed

Breathing-kagome niobium halides

Flat-band cluster Mott insulators and the Nb₃Br₈ Josephson diode

abs:Nb3Cl8 OR abs:Nb3Br8 OR abs:Nb3I8

Open the live feed

Weyl and Dirac semimetal tellurides

Room-temperature nonlinear Hall effects and a dual quantum spin Hall insulator in TaIrTe₄, and topology on a knife edge in the pentatellurides

abs:TaIrTe4 OR abs:NbIrTe4 OR abs:ZrTe5 OR abs:HfTe5

Open the live feed

Quasi-one-dimensional trichalcogenides

Sliding charge-density waves in chain metals, and anisotropic semiconductors and wires built from layers of atomic chains

abs:TiS3 OR abs:ZrS3 OR abs:ZrTe3 OR abs:NbS3 OR abs:NbSe3 OR abs:TaSe3 OR abs:TaS3

Open the live feed

Trigonal PtBi₂

A layered Weyl semimetal whose Fermi-arc surface states superconduct with a nodal gap while the bulk stays normal

abs:PtBi2

Open the live feed

Tantalum nickel selenide

The leading excitonic-insulator candidate, where electron-hole pairing and a lattice distortion arrive together

abs:Ta2NiSe5 OR abs:Ta2NiS5

Open the live feed

Kagome metals AV₃Sb₅

Charge order, possible time-reversal symmetry breaking and superconductivity on a vanadium kagome lattice

abs:CsV3Sb5 OR abs:KV3Sb5 OR abs:RbV3Sb5 OR abs:AV3Sb5

Open the live feed

MoSi₂N₄ and the MA₂Z₄ family

A 2D family built by design rather than exfoliation, mostly predicted and only partly grown

abs:MoSi2N4 OR abs:WSi2N4 OR abs:MA2Z4

Open the live feed

Nitride halides ZrNCl and HfNCl

Superconductivity in a doped band insulator, where the transition temperature rises as carriers are removed

abs:ZrNCl OR abs:HfNCl

Open the live feed

Latest items

Newest papers and preprints tagged for this track, from the news feed updated 5 Oct 2026.

Preprintnot yet peer reviewed 5 Oct 2026 arXiv

Quantum materials QED with van der Waals crystals

A central goal of the emerging field of materials QED is to harness subwavelength electromagnetic confinement in engineered cavities to tailor light-matter interactions. Here, we demonstrate that van der Waals multilayer cavities composed of stacked graphene and hexagonal boron nitride (hBN) provide unprecedented…

TheoryGraphene hBN

Preprintnot yet peer reviewed 5 Oct 2026 arXiv

Electrostatic Doping of Moiré Superlattices Controls the Optical Fingerprint of a WSe_2 /Twisted Bilayer Graphene heterostructure

We theoretically investigate the optical response of the WSe2 monolayer vertically stacked on twisted bilayer graphene (tBG) under electrostatic doping. In this heterostructure, the doped moiré superlattice of tBG generates a spatially modulated electrostatic potential that couples to the electron and hole constituents…

TheoryTwisted graphene Graphene WSe₂

Preprintnot yet peer reviewed 2 Oct 2026 arXiv

Electrostatic Doping of Moiré Superlattices Controls the Optical Fingerprint of a WSe_2 /Twisted Bilayer Graphene heterostructure

TheoryExperimentTwisted graphene Graphene WSe₂

Preprintnot yet peer reviewed 1 Oct 2026 arXiv

Parafermions in fractional Chern insulator-superconductor heterostructures: the role of spin polarization

Most proposals for Z₃ parafermions in fractional quantum Hall-superconductor structures used the spin-unpolarized ν= 2/3 Halperin (1,1,2) state. The (FQAH) states of twisted MoTe₂ and rhombohedral graphene are believed to be spin- and valley-polarized Jain states, with the same…

TheoryTwisted graphene Graphene MoTe₂

Preprintnot yet peer reviewed 1 Oct 2026 arXiv

Large-gap quantum anomalous Hall insulator in two-dimensional pentagonal TiN₈ monolayer

Discovering the quantum anomalous Hall (QAH) effect in two-dimensional magnets is a central goal of topological condensed matter physics, yet candidate materials combining a sizable gap with a well-defined Chern number remain scarce. We report a new material platform that realizes emergent magnetism and nontrivial band…

Theory

All 961 items in the news feed

Conferences

The next meetings where this track’s work is presented, with dates checked against each organiser’s own site.

16–20 Nov 2026 Sydney, Australia

RPGR 2026

Abstract submission closed; registration open

Specialist meetingTheoryExperimentEngineering

29 Nov – 4 Dec 2026 Boston, USA

2026 MRS Fall Meeting & Exhibit

Breaking-news abstracts are being accepted; the deadline is on the organiser’s site

Society meetingTheoryExperimentEngineering

6–12 Mar 2027 Kirchberg in Tirol, Austria

IWEPNM 2027

The call for abstracts opens in early autumn

SchoolTheoryExperiment

14–19 Mar 2027 Regensburg, Germany

DPG Spring Meeting 2027

Abstracts close 1 Dec 2026 · in 57 days

Society meetingTheoryExperiment

All 11 upcoming conferences

Reading list

13 papers worth reading in full, and why each one is here.

The electronic properties of graphene Castro Neto et al. · Reviews of Modern Physics 81, 109 (2009) cited by 24,886 Still the standard reference for Dirac physics, disorder and interactions in graphene.
Quantum spin Hall effect in graphene Kane & Mele · Physical Review Letters 95, 226801 (2005) cited by 8,173 The model that launched Z₂ topology; every heavy-Xene and WTe₂ paper builds on it.
Coupled spin and valley physics in monolayers of MoS₂ and other group-VI dichalcogenides Xiao et al. · Physical Review Letters 108, 196802 (2012) cited by 5,166 The minimal massive-Dirac model behind valley selection rules and spin–valley locking.
Moiré bands in twisted double-layer graphene Bistritzer & MacDonald · PNAS 108, 12233 (2011) cited by 3,100 The continuum model that predicted magic angles seven years before experiment.
Van der Waals heterostructures Geim & Grigorieva · Nature 499, 419 (2013) cited by 10,947 The conceptual framing of 2D crystals as stackable building blocks.
Colloquium: Excitons in atomically thin transition metal dichalcogenides Wang et al. · Reviews of Modern Physics 90, 021001 (2018) cited by 2,133 A thorough theory-and-experiment account of excitons, screening and valley physics.
Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models Mermin & Wagner · Physical Review Letters 17, 1133 (1966) cited by 8,354 Know precisely what the theorem forbids – and why anisotropy is the loophole every 2D magnet uses.
van der Waals bonding in layered compounds from advanced density-functional first-principles calculations Björkman et al. · Physical Review Letters 108, 235502 (2012) cited by 1,163 RPA benchmarks of interlayer binding across layered compounds – a reference point for choosing dispersion corrections.
Dielectric genome of van der Waals heterostructures Andersen, Latini & Thygesen · Nano Letters 15, 4616 (2015) cited by 230 A practical way to include the screening of whole heterostructures in quasiparticle and exciton calculations.
Charged point defects in the flatland: accurate formation energy calculations in two-dimensional materials Komsa et al. · Physical Review X 4, 031044 (2014) cited by 115 Why 3D charge corrections fail in slabs, and how to do defect energetics in 2D properly.
First-principles calculations of charge carrier mobility and conductivity in bulk semiconductors and two-dimensional materials Poncé et al. · Reports on Progress in Physics 83, 036501 (2020) cited by 356 A careful guide to ab initio transport, including the 2D-specific pitfalls.
Two-dimensional materials from high-throughput computational exfoliation of experimentally known compounds Mounet et al. · Nature Nanotechnology 13, 246 (2018) cited by 1,875 The screen behind the ‘over a thousand exfoliable compounds’ figure, and a model for reproducible high-throughput work.
Magic-angle twisted bilayer graphene as a topological heavy fermion problem Song & Bernevig · Physical Review Letters 129, 047601 (2022) cited by 258 A reframing of magic-angle graphene that connects moiré physics to heavy-fermion theory.

Glossary

49 terms this track leans on. Each one opens its entry, which explains it in plain words and for specialists.

The whole glossary, 204 terms

Where next

BasicsWhat is a 2D material?What they are, why being thin changes everything, and how they are made. CourseLearn it step by step10 levels, from atoms to the frontier, with questions to check yourself. Catalogue74 material familiesWhat each one is, how it is made, and how close it is to real use. Applications18 areas, sorted by readinessFrom composites already sold by the tonne to physics with no product yet. NewsThe latest papers and storiesCollected from 137 feeds and tagged by material and audience.

Strong electron correlation

What happens when electrons in a material repel each other so strongly that they can no longer be treated as moving independently. Each electron’s motion then depends on where all the others are, and the result can be an insulator where simple theory predicts a metal, or magnetism, superconductivity and other collective states.

The regime in which the Coulomb repulsion U between electrons is comparable to or larger than their kinetic energy, set by the bandwidth W, so single-particle band theory fails. At half filling with U well above W the Hubbard model gives a Mott insulator with local moments and antiferromagnetic exchange of order t²/U; doping or tuning U/W gives correlated metals, heavy-fermion behaviour and unconventional superconductivity. In 2D, weak screening keeps U large and moiré superlattices shrink W, so twisted bilayers make U/W tunable by gate and twist. Semi-local DFT misses the physics; DFT+U, DMFT, exact diagonalisation and quantum Monte Carlo are the usual tools.

Full explanation

Gating

Changing a material by putting a voltage on a nearby electrode – a gate – separated from it by an insulator. The voltage pulls electrons into the material or pushes them out, as in every transistor. Because a 2D material is so thin, the gate reaches all of it, so the number of electrons can be dialled up and down continuously: a single device can be turned from insulator to metal, its magnetism strengthened, or superconductivity switched on. Two gates, above and below, can also apply an electric field across the layer.

Electrostatic control of carrier density and electric field through a gate coupled capacitively across a dielectric: the induced density equals the gate capacitance times the voltage beyond threshold, divided by the electron charge, reaching a few 10¹³ cm⁻² with oxide or hBN gates before breakdown. Dual gating sets density and perpendicular displacement field independently, opening the gap of bilayer graphene and tuning moiré flat bands. Ionic-liquid and solid-electrolyte gating form an electric double layer about a nanometre thick and reach 10¹⁴–10¹⁵ cm⁻², enough to induce superconductivity in MoS₂ or raise the Curie temperature of Fe₃GeTe₂, but can intercalate or react electrochemically and work only while the ions are mobile. Hysteresis from traps, contact effects and quantum capacitance are the usual caveats.

Full explanation

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

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

Disorder and localisation

Every real crystal has some disorder: missing or misplaced atoms, impurities, stray charges in the substrate, a random mixture of elements in an alloy. Electrons moving through it scatter off these imperfections, which raises the resistance. If the disorder is strong enough, electron waves become trapped in small regions – localised – and the material stops conducting, even if it would be a metal when perfect. In a sheet one atom thick, disorder in its surroundings matters especially.

Deviations from perfect periodicity – point defects, substitutional or alloy disorder, interface roughness, charged impurities and strain variations in the substrate – that scatter carriers and, beyond a threshold, localise their wavefunctions through interference (Anderson localisation). Scaling theory predicts that non-interacting electrons in two dimensions without spin–orbit coupling are localised by any disorder; its precursor, weak localisation, appears as a logarithmic rise of resistance at low temperature that a magnetic field suppresses, while strong spin–orbit coupling gives weak antilocalisation. In graphene on SiO₂, charged impurities break the sheet near charge neutrality into electron–hole puddles, which hBN encapsulation reduces. Strong disorder leads to hopping conduction, broadens optical lines and smears phase transitions; in frustrated magnets and correlated oxides, structural or cation disorder can mimic exotic states.

Full explanation

Hall effect

Send a current along a strip in a magnetic field and the moving charges are pushed towards one edge, so a small voltage appears across the strip. Its size tells how many charge carriers there are, and its sign whether they are electrons or holes, which makes it the standard way to count them. In a magnetic material a sideways voltage appears even without an outside field – the anomalous Hall effect, a handy way to see whether a tiny flake is magnetic.

The transverse voltage that the Lorentz force produces when a current flows in a perpendicular magnetic field. For one type of carrier in a 2D sheet the Hall resistance is B/ne, independent of thickness, so its slope gives the sheet density n and its sign the carrier type; combined with the sheet resistance it gives the Hall mobility. Two carrier types make the Hall resistance non-linear in B and call for a two-band fit. In magnetic conductors an extra term that follows the magnetisation – the anomalous Hall effect – comes from the Berry curvature of the bands (intrinsic) or from skew and side-jump scattering (extrinsic); quantised, it becomes the quantum anomalous Hall effect.

Full explanation

Quantum Hall effect

In a strong magnetic field at low temperature, electrons flowing along a flat sheet are pushed to one side, so a voltage builds up across it. That sideways voltage, divided by the current, stops changing smoothly and locks onto exact steps set only by fundamental constants – the same in every sample, and so precise that they are used to define the ohm.

Quantisation of the Hall conductance in units of e²/h when a 2D electron system sits in a strong perpendicular field and the Fermi level lies between Landau levels, with the longitudinal resistance vanishing. Graphene shows a half-integer sequence with a level pinned at zero energy, a direct consequence of its Dirac dispersion and π Berry phase, and the effect survives to room temperature there. The zero-field analogues are the quantum anomalous and quantum spin Hall effects.

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

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

Exchange interaction

The force that lines up the tiny magnets of neighbouring atoms in a magnetic material – all the same way in a ferromagnet, alternating in an antiferromagnet. Despite what it does, it is not magnetic in origin: it comes from the electric repulsion between electrons combined with a quantum rule that keeps two electrons from sharing the same state, which makes their energy depend on whether their spins point the same way. It is far stronger than the magnetic pull between atomic magnets, which is why iron stays magnetic up to 770 °C.

The spin-dependent energy that arises from the Coulomb repulsion together with the antisymmetry of the many-electron wavefunction, modelled as a coupling J between neighbouring spins in Heisenberg, Ising or XY form. In insulators it is mostly mediated by ligands – superexchange, whose sign follows the Goodenough–Kanamori rules: antiferromagnetic for metal–ligand–metal angles near 180°, often ferromagnetic near 90°, as in the edge-sharing octahedra of CrI₃ – in metals by conduction electrons (RKKY, double exchange), and with strong spin–orbit coupling it becomes anisotropic or bond-dependent (Kitaev). In van der Waals magnets the intralayer exchange of a few meV sets the ordering scale, while a much weaker interlayer exchange decides ferro- or antiferromagnetic stacking and depends on the stacking itself; ordering a monolayer additionally requires anisotropy, by the Mermin–Wagner theorem.

Full explanation

Topological phase

A state of matter told apart not by how its atoms are arranged but by a whole number that describes how its electrons’ waves twist across the crystal – the way a doughnut differs from a ball by its one hole. A whole number cannot change a little, so what it guarantees, such as current running along an edge without loss, survives defects and dirt until the band gap itself closes.

A gapped phase characterised by a topological invariant of its occupied bands – a Chern number, a Z₂ index, a winding number – that cannot change under deformations that keep the gap open and any protecting symmetry intact. Where regions with different invariants meet, the gap must close, which forces boundary states: chiral edge channels in Chern insulators, helical ones in quantum spin Hall insulators, Fermi arcs in Weyl semimetals, Majorana modes in topological superconductors. Topological order in the strict sense – the long-range entanglement of fractional quantum Hall states and spin liquids – is a distinct, stronger notion.

Full explanation

Spin

A built-in property of every electron that makes it a tiny magnet. Measure it along any direction and you only ever find one of two answers, ‘up’ or ‘down’. Despite the name, nothing is actually spinning – the word stuck from an early picture – but the magnetism is real: countless electron spins lined up are what make a fridge magnet stick. Spintronics tries to carry information in spin rather than in charge.

The intrinsic angular momentum of the electron, ħ/2, with a magnetic moment of almost exactly one Bohr magneton; a measurement along any axis yields one of two values. In solids spin couples to orbital motion through spin–orbit coupling and to other spins through exchange; in 2D materials it sets magnetic order, spin–valley locking and how long a spin survives as a carrier of information.

Full explanation

Neutron scattering

Shooting neutrons at a crystal and watching how they bounce off. Neutrons carry no charge, so they pass deep into a sample, but they carry a tiny magnetic moment, so they feel the magnetic moments of atoms. That makes them the standard tool for finding how the spins in a magnet are arranged and – from the energy the neutrons lose – how the spins wave and wobble. The catch is that neutrons come only from research reactors and large accelerators, and the samples have to be large crystals.

Elastic and inelastic scattering of thermal and cold neutrons from nuclei and, through the neutron’s magnetic moment, from unpaired electrons. Diffraction gives the nuclear structure, with sensitivity to light elements and isotopes, and the magnetic structure from magnetic Bragg peaks below the ordering temperature (Shull, 1949); polarised neutrons separate magnetic from nuclear scattering. Inelastic scattering on triple-axis and time-of-flight spectrometers maps phonon and magnon dispersions, exchange constants and continua – the magnon gaps of CrI₃, the scattering continuum of α-RuCl₃. Weak fluxes require gram-scale samples or many co-aligned crystals, so 2D materials are measured in bulk form; monolayers are out of reach, and bulk results are transferred to thin layers with care.

Full explanation

Interlayer coupling

How strongly neighbouring sheets in a stack affect one another. The pull holding them together is weak, but electrons can still hop from one sheet to the next and whole sheets can vibrate against each other – which is why one, two and many layers of the same crystal behave differently, and why twisting or sliding one sheet changes the whole stack.

The electronic hybridisation, mechanical force constants and electrostatic interaction between adjacent layers of a van der Waals crystal. Electronically it is hopping between orbitals that reach out of the plane, a few tenths of an electronvolt in graphite, which splits and shifts bands and makes the band structure depend on thickness, stacking and twist. Mechanically it sets the interlayer shear and breathing modes, whose Raman frequencies of a few tens of cm⁻¹ measure it directly. Far weaker than in-plane bonding, it still decides the direct–indirect gap crossover of TMDCs, the flat bands of twisted bilayers and interlayer magnetic order.

Full explanation

Ferroelectricity

A material that holds an electric polarisation – one side slightly positive, the other slightly negative – with no voltage applied, and that can be flipped by a field and stay flipped. That memory of the last field applied is what makes it useful for storing data.

A spontaneous, switchable electric polarisation produced by a structural distortion that breaks inversion symmetry. In 2D the usual obstacle – depolarising fields destroying the polarisation as a film thins – is sidestepped in three ways: in-plane polarisation, layered compounds whose dipoles survive to the monolayer such as CuInP₂S₆ and In₂Se₃, and sliding ferroelectricity, where the polarisation belongs to the interface between layers rather than to the layers themselves.

Full explanation

Janus 2D material

A single layer whose two faces are made of different atoms – selenium on top and sulfur underneath, for example – named after the two-faced Roman god. Because top and bottom differ, the sheet carries a built-in electric field from one face to the other, which a symmetric layer cannot have. That lopsidedness allows effects a symmetric sheet forbids, such as producing a voltage when it is squeezed from above.

A monolayer in which the two outer atomic planes are different elements, as in MoSSe, where one chalcogen sheet is S and the other Se. Mirror symmetry through the metal plane is broken (D₃h → C₃v for the 1H structure), giving an intrinsic out-of-plane dipole, out-of-plane piezoelectricity, Rashba-type spin splitting and an out-of-plane second-harmonic response. Most are made by replacing one face of a parent monolayer; RhSeCl is a rare bulk crystal whose layers are Janus by nature.

Full explanation

Metal, semimetal and insulator

The three basic ways a solid treats electricity, set by how its electrons fill the energies open to them. In a metal they fill a band of energies only partly, so the smallest push sets them moving, and it conducts at any temperature. In an insulator they fill a band completely and a wide gap separates it from the next, so nothing can move. A semimetal sits in between: two bands just touch or overlap a little, leaving a small number of mobile electrons and holes. NbSe₂ is a metal, graphene a semimetal, hBN an insulator.

A classification by where the Fermi level falls in the band structure. In a metal it crosses a partly filled band, leaving a Fermi surface, a carrier density of order one per atom (about 10²² cm⁻³) and a resistivity that falls on cooling. In a band insulator it lies in a gap of several electronvolts; a semiconductor is an insulator whose gap is small enough to be doped or gated. A semimetal has a slight overlap between conduction and valence bands (graphite, bismuth, bulk WTe₂) or bands that touch at isolated points (graphene), giving low and nearly equal densities of electrons and holes. Interactions can open gaps that band theory does not predict: Mott insulators from half-filled bands, excitonic insulators from semimetals.

Full explanation

Edge state

Electrons that can only travel along the border of a sheet while its interior stays insulating. In a topological material these border lanes are guaranteed to exist, and in the cleanest cases they carry current without losing energy to scattering.

States localised at a sample boundary and dispersing inside the bulk gap. In quantum Hall and Chern insulators they are chiral, one-way channels; in a quantum spin Hall insulator they are helical, counter-propagating with opposite spins and protected by time reversal, giving e²/h per edge in the ideal case – monolayer 1T′-WTe₂ keeps that edge conduction up to about 100 K. Trivial edges carry states too, from dangling bonds, reconstructions and graphene’s zigzag edge, so the evidence for topology is quantised conductance and its magnetic-field dependence, not conduction at the edge alone.

Full explanation

Excitonic insulator

A state in which electrons and the holes they leave behind bind into pairs so strongly that the material opens a gap and stops conducting – not because of its chemistry, but because the pairs have all settled together into one shared, collective state. Telling this apart from an ordinary distortion of the crystal is notoriously difficult.

A ground state predicted for a semimetal with small band overlap, or a semiconductor whose gap is smaller than the exciton binding energy, in which spontaneous exciton condensation reconstructs the bands and opens a gap. The experimental difficulty is that the transition is usually accompanied by a lattice distortion of the same symmetry, so photoemission, ultrafast and pressure experiments have to separate an electronic condensate from a conventional Peierls-like instability – the argument that surrounds Ta₂NiSe₅ and 1T-TiSe₂.

Full explanation

Berezinskii–Kosterlitz–Thouless transition

How a flat, two-dimensional system that cannot have perfect order can still switch between an almost-ordered state and a disordered one. At low temperature, tiny whirlpools in the pattern of its spins – or of a superconductor – exist only in tightly bound pairs; above a critical temperature they break free and scramble the order.

A phase transition without a local order parameter in two-dimensional systems with a continuous planar symmetry – XY magnets, superfluid and superconducting films – driven by the unbinding of vortex–antivortex pairs. Below T_BKT correlations decay as a power of distance (quasi-long-range order), above it exponentially. The superfluid stiffness jumps to zero from the universal value 2k_BT_BKT/π, and in superconducting films the current–voltage curve follows V ∝ I³ at T_BKT. It evades the Mermin–Wagner theorem, which forbids true long-range order but not this.

Full explanation

Point defect

A flaw in a crystal that involves a single atomic site: an atom missing (a vacancy), an atom of the wrong kind in a site (a substitution, or an antisite when the crystal’s own two kinds of atom swap places), or an extra atom squeezed in between (an interstitial). Every real crystal has some. In a sheet a few atoms thick each one sits at the surface, so a handful of them can change how the material conducts, glows or reacts far more than in a thick crystal.

A defect confined to one or a few lattice sites – vacancy, interstitial, substitutional impurity, antisite or adatom – or a small complex of them. What it does is set by the levels it introduces: shallow levels dope, deep levels trap carriers and act as non-radiative recombination centres, and an isolated deep level in a wide gap can emit single photons. Formation energies and charge states depend on the chemical conditions of growth and on the Fermi level. In 2D materials every defect is a surface defect, less screened and more exposed to the environment, and densities of 10¹² to 10¹³ per cm² are typical of TMDC monolayers.

Full explanation

Electron–phonon coupling

The push and pull between a crystal’s electrons and the vibrations of its atoms. Vibrating atoms scatter moving electrons and slow them down, which is why a metal conducts worse when warm; the electrons in turn shake the atoms and lose energy to them as heat. The same coupling can also draw electrons together: it is the glue that pairs them in ordinary superconductors, and when it is strong it can make the whole crystal buckle into a charge density wave.

The change in electronic energies when atoms are displaced, described by matrix elements between electron states linked by a phonon and summarised by the dimensionless constant λ obtained from the Eliashberg function α²F(ω). It limits room-temperature mobility through phonon scattering, gives metals their linear-in-temperature resistivity at high temperature, renormalises quasiparticle bands (the kinks seen in ARPES), mediates the attraction behind conventional superconductivity and, where it is strong at particular wavevectors, softens phonons into a charge density wave. In 2D materials it is computed from first principles with density functional perturbation theory and Wannier interpolation, and it can be tuned by doping, strain and the dielectric environment.

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

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 alignment

How the energy levels of two materials line up when they are put together: whether one material’s electrons sit above or below the other’s. Electrons roll ‘downhill’ to the lower level, so the alignment decides which way charge moves across the junction, and so decides what the stack can be used for.

The relative positions of the band edges of two materials in contact, classified as straddling (type I), staggered (type II) or broken (type III). Van der Waals stacking avoids chemical bonding and strain, so alignments start close to the isolated layers’ ionisation potentials and electron affinities, shifted by interface dipoles and by the dielectric environment. A type-II alignment puts electron and hole in different layers, which is what makes interlayer excitons and gate-tunable photovoltaic junctions possible.

Full explanation

Molecular dynamics and machine-learned potentials

Molecular dynamics is a computer simulation that follows every atom in a material through time, like the frames of a film: work out the force on each atom, nudge it a tiny step, repeat millions of times. It shows how a crystal melts, cracks or carries heat. The hard part is the forces. Exact quantum calculations are slow, so machine-learned potentials are trained on a few thousand of them and then reproduce the forces far faster.

Molecular dynamics integrates Newton’s equations for all atoms with a time step of about a femtosecond, giving trajectories from which structure, diffusion, thermal conductivity and phase behaviour follow. Forces come from empirical force fields, from DFT at every step (ab initio MD, limited to hundreds of atoms and picoseconds), or from machine-learned interatomic potentials – neural-network, kernel or equivariant graph models fitted to DFT energies and forces – that approach DFT accuracy at a small fraction of the cost within the configurations they were trained on.

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.

Full explanation

Mirror symmetry

A crystal has a mirror symmetry if reflecting it in a plane leaves it unchanged. For a 2D sheet the question that matters most is whether it looks the same from above and from below – which decides how its electrons’ spins behave, which band crossings are protected, and even how a calculation of it has to be set up.

Invariance under reflection through a plane. For a layer the horizontal mirror in the plane of the sheet matters most: it forbids an out-of-plane dipole, so a slab without it needs a dipole correction in a periodic calculation, and in a 2H monolayer that also lacks inversion it keeps spin–orbit-split spins pointing out of plane, the origin of Ising pairing. Vertical mirrors can protect band crossings – mirror-protected nodal lines in PbTaSe₂, mirror Chern numbers in the SnTe class of topological crystalline insulators – and losing a mirror at a structural transition is how some chain compounds become ferroelectric.

Full explanation

Fermi surface and nesting

The Fermi surface is the boundary between the states a metal’s electrons fill and the ones they leave empty, drawn on a map of every way an electron can move through the crystal; its shape decides how the metal conducts. Nesting is when large flat stretches of that boundary can be slid onto each other by one single shift – a metal like that is unstable and tends to settle into a regular ripple of charge or spin.

The constant-energy surface separating occupied from empty states, whose geometry governs transport, screening and instabilities. Nesting – parallel sheets connected by a single wavevector q – peaks the bare susceptibility at q and is the textbook route to charge and spin density waves. In real layered metals the ordering wavevector often follows momentum-dependent electron–phonon coupling instead, so a nesting figure alone does not settle the mechanism.

Full explanation

Chalcogenide, halide and pnictide

Family names for compounds, taken from the element in them that carries the negative charge. Chalcogenides contain sulfur, selenium or tellurium; halides contain fluorine, chlorine, bromine or iodine; pnictides contain phosphorus, arsenic, antimony or bismuth. Most layered crystals belong to one of these families, and the catalogue groups many of them that way.

Compounds named for their most electronegative constituent: chalcogenides of group 16 (S, Se, Te; oxides are conventionally counted apart), halides of group 17 (F, Cl, Br, I) and pnictides of group 15 (P, As, Sb, Bi; nitrides usually keep their own name). Down each group the anion grows larger and less electronegative, so bonding turns more covalent, gaps shrink and spin–orbit coupling grows. Chalcogenides give the MX₂ semiconductors and metals, halides mostly ionic, often magnetic insulators built from edge-sharing octahedra, and pnictides many metals, semimetals and the parents of the iron-based superconductors; mixed-anion compounds such as oxychalcogenides and chalcohalides combine two.

Full explanation

Commensurate and incommensurate

Whether two repeating patterns fit each other. If one pattern repeats after exactly a whole number of steps of the other – every three atoms, say – the two are commensurate, and together they repeat again after a short distance. If the ratio is not a simple fraction, they are incommensurate and never line up exactly. The question comes up whenever a crystal grows a second pattern of its own – a charge density wave, a magnetic spiral – and whenever two different layers are stacked, as in a moiré.

A modulation is commensurate when its wavevector is a rational fraction of a reciprocal lattice vector, so that crystal and modulation share a finite supercell, and incommensurate when the ratio is irrational, leaving no exact common period. The charge density wave of 1T-TaS₂ passes from incommensurate (below about 550 K) to nearly commensurate – commensurate domains separated by discommensurations – and locks into the commensurate √13 × √13 Star-of-David phase below about 180 K; RTe₃ keeps an incommensurate wave, and NiI₂ an incommensurate spin spiral. For two lattices – graphene on hBN, twisted bilayers – the same competition between elastic energy and interlayer adhesion decides whether the layers stretch into commensurate domains or keep their own lattices.

Full explanation

Magnetoresistance

A change in electrical resistance when a magnetic field is applied. It is how the read head of a hard disk senses its bits, and in some layered magnets the change is enormous – a stack can go from barely conducting to conducting – which is how the magnetic state of a few atomic layers is read out electrically.

The field dependence of resistivity. Ordinary orbital magnetoresistance is positive and usually small, though enormous in compensated semimetals such as WTe₂; giant and tunnelling magnetoresistance come from spin-dependent transport across magnetic layers; colossal magnetoresistance from a field-driven change of electronic state. In van der Waals stacks a few layers of CrI₃ act as a spin filter whose resistance changes by orders of magnitude as adjacent layers align, which is how layer-by-layer magnetic order became measurable in transport.

Full explanation

Complementary metal–oxide–semiconductor (CMOS)

The way almost every digital chip is built. Each logic gate pairs a transistor that conducts with electrons with one that conducts with holes, so one of the two is always off and the gate draws almost no current except while it switches. A 2D material becomes useful for logic only if both kinds of transistor can be made from it, or if it can be added to a silicon chip without harming it.

Logic in which n- and p-channel field-effect transistors are wired in series between supply and ground, so that in either stable state one of them is off and static power is set by leakage alone. It demands n- and p-type devices of comparable drive, threshold voltages held within tens of millivolts across a wafer, and a process compatible with the silicon line – about 400 °C at most for back-end-of-line integration. For 2D channels the p-type half is the harder one: most TMDCs conduct n-type with common contact metals, and WSe₂ is the leading p-type candidate.

Full explanation