E′ and A′₁ modes

In plain words

Two characteristic vibrations of a of MoS₂ and its relatives, one along the sheet and one out of it. Each shows up as a peak in the spectrum, and the gap between the two peaks widens as layers are added – a quick way to count them.

Going deeper

These are the two Raman modes that dominate a TMDC spectrum: one motion within the plane, one across it. Adding layers softens the in-plane mode and stiffens the out-of-plane one, so the distance between the peaks grows – the standard quick way to count layers.

What the atoms do

In the E′ mode the metal atom and the two move in opposite directions within the plane of the layer; in the A′₁ mode the chalcogens move up and down along the perpendicular direction while the metal stays still. In monolayer MoS₂ the first sits near 385 cm⁻¹ and the second near 404 cm⁻¹.

The primes in the labels come from the symmetry of a single layer, which has a plane but no . The bulk crystal has a centre of inversion, so the same vibrations are labelled E¹₂g and A₁g there (Eg and A₁g in even layer numbers) – the reason the literature uses both sets of names for what is essentially the same pair of motions.

Why their separation counts layers

Adding a second layer changes the two modes in opposite directions. The out-of-plane mode stiffens, because the neighbouring layer resists the chalcogens’ motion through interaction. The in-plane mode softens slightly, which is usually attributed to the increased of long-range interactions between the charges on the atoms. The separation therefore grows monotonically: about 19 cm⁻¹ for a monolayer of MoS₂, rising towards 25 cm⁻¹ in the bulk.

Because it is a difference between two peaks in the same spectrum, the measure is insensitive to calibration offsets, which is what makes it so widely used. The steps become small beyond about four layers, so the method counts thin samples well and thick ones poorly.

What else moves the peaks

shifts the in-plane mode strongly and can split it, since it responds to stretching within the plane; shifts the out-of-plane mode, which couples to electrons more strongly, and broadens it. Temperature shifts both. So a layer count taken from the separation assumes an unstrained, undoped, room-temperature sample on a standard – and where those assumptions fail, the same measurement becomes a probe of strain and doping instead.

Two more practical points: the numbers differ between materials, so MoS₂ values must not be used for WS₂ or MoSe₂; and low-frequency shear and , below about 50 cm⁻¹, count layers far more sharply, because they exist only when there is more than one layer – at the cost of needing a spectrometer able to reach close to the laser line.

For specialists

The in-plane and out-of-plane Raman-active vibrations of 1H monolayers, whose separation is a common layer-number indicator.

Where this comes from

Anomalous lattice vibrations of single- and few-layer MoS₂ Lee et al. · ACS Nano 4, 2695 (2010) cited by 4,882
Phonons in single-layer and few-layer MoS₂ and WS₂ Molina-Sánchez and Wirtz · Physical Review B 84, 155413 (2011) cited by 1,442
Phonon and Raman scattering of two-dimensional transition metal dichalcogenides from monolayer, multilayer to bulk material Zhang et al. · Chemical Society Reviews 44, 2757 (2015) cited by 1,309

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

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

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

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

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