Electron–phonon coupling

In plain words

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 , and when it is strong it can make the whole crystal buckle into a .

Going deeper

One coupling, three consequences. Atoms jiggling with heat scatter electrons and cap their mobility; a passing electron pulls the lattice in behind it and the positive wake draws a second electron along, pairing them in a conventional superconductor; and where the coupling is strong enough, the atoms shift for good and the electron density ripples with them in a charge density wave.

What sets the speed limit

In a clean sample at room temperature, are the obstacle that cannot be removed: impurities can be avoided by better growth, lattice vibrations cannot. How strongly they couple to the carriers therefore sets an intrinsic ceiling on . For graphene it is about 200,000 cm²/Vs, far above silicon; for MoS₂ calculations give about 400 cm²/Vs at room temperature, and measured values sit below that because of charged impurities, defects and the .

The substrate can add phonons of its own. Polar substrates such as SiO₂ carry surface vibrations whose electric fields reach into the 2D layer and scatter its carriers – one reason why graphene on SiO₂ stays far below its intrinsic limit, and why hBN, whose own polar vibrations lie higher in energy and are less excited at room temperature, makes a better neighbour.

The glue in superconductors

Exchanging a phonon lets two electrons attract each other despite their mutual repulsion: a passing electron pulls the positive ions together, and the slight excess of positive charge it leaves behind draws in a second electron a moment later. That attraction, at the heart of the theory of Bardeen, Cooper and Schrieffer of 1957, binds electrons into the pairs of a conventional superconductor, and the rises with λ and with the energy of the phonons involved.

In layered materials it is the mechanism behind superconductivity in NbSe₂ and in graphite with calcium, and calculations of λ guide the search for new examples. When the computed coupling is far too weak for the measured critical temperature, as in the iron-based superconductors, the mismatch is itself evidence that something else does the pairing.

When the lattice gives way

If the coupling is strong for one particular wavelength, the vibration with that wavelength softens as the crystal cools – a dip in its frequency known as a Kohn anomaly – until it freezes in as a static distortion with a modulated electron density: a charge density wave. In NbSe₂ the softening spreads over a wide range of wavevectors, which showed that the momentum dependence of the coupling, not the shape of the alone, chooses the pattern.

Strong coupling to a local distortion can also trap a carrier in the dent it makes in the lattice, forming a , as in the soft lattices of layered perovskites. Optical spectra show the same coupling as phonon side bands and broadened lines, measured by the .

For specialists

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 at high temperature, renormalises bands (the kinks seen in ), mediates the attraction behind conventional superconductivity and, where it is strong at particular wavevectors, softens phonons into a charge density wave. In it is computed from with density functional perturbation theory and Wannier interpolation, and it can be tuned by , and the .

Where this comes from

Intrinsic and extrinsic performance limits of graphene devices on SiO₂ Chen et al. · Nature Nanotechnology 3, 206 (2008)
Phonon-limited mobility in n-type single-layer MoS₂ from first principles Kaasbjerg et al. · Physical Review B 85, 115317 (2012)
Extended phonon collapse and the origin of the charge-density wave in 2H-NbSe₂ Weber et al. · Physical Review Letters 107, 107403 (2011)
Electron-phonon interactions from first principles Giustino · Reviews of Modern Physics 89, 015003 (2017)

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

Phase transition

A sudden change in the state of a material when temperature, pressure or another control is varied – ice melting into water, a magnet losing its magnetism when heated, a metal becoming a superconductor when cooled. Each happens at a definite transition temperature. In quantum materials, transitions mark where new kinds of order appear: charge density waves, magnetism, superconductivity, ferroelectricity. A map of which state appears where, against temperature and a second knob such as doping or pressure, is a phase diagram.

A non-analytic change of thermodynamic state at a transition point, described by an order parameter that vanishes in the disordered phase and is finite in the ordered one – magnetisation, polarisation, charge-density-wave amplitude, superconducting gap. First-order transitions show latent heat, phase coexistence and hysteresis; continuous ones show a diverging correlation length and susceptibility and universal critical exponents set by dimension and symmetry (Landau theory, the renormalisation group). In two dimensions fluctuations are stronger: the Mermin–Wagner theorem forbids breaking a continuous symmetry at finite temperature, leaving anisotropic, Ising-like order or Berezinskii–Kosterlitz–Thouless quasi-order. Transition temperatures change with layer number, gating, strain and twist, which makes 2D materials tunable platforms for phase diagrams, including quantum phase transitions driven at zero temperature by non-thermal parameters.

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

Resistivity and sheet resistance

Two ways to put a number on how strongly a material resists a current. Resistivity belongs to the material itself, whatever the size of the piece: copper’s is tiny, glass’s enormous. For a sheet so thin that its thickness is hard to pin down, the sheet resistance is used instead: the resistance of a square of it, measured from one edge to the opposite one, which comes out the same for a square of any size. It is quoted in ohms per square.

Resistivity ρ relates electric field to current density, in Ω·m or Ω·cm, so that a bar of length L and cross-section A has resistance ρL/A. For a film of thickness t the sheet resistance is ρ/t, quoted in Ω per square, and a strip L long and W wide has the sheet resistance times L/W. For an atomically thin layer it is the natural quantity, since the thickness is a convention, and it equals 1/neμ for sheet density n and mobility μ. Both are measured with four probes – a Hall bar, a collinear four-point probe or the van der Pauw method for arbitrary shapes – so that the voltage is sensed away from the current contacts and the contact resistance drops out.

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

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