A stripped-down model in which electrons sit on atoms and hop to their neighbours with a fixed probability. It predicts little on its own, but once its handful of numbers is fitted it reproduces bands cheaply enough to handle millions of atoms – a , for instance.
Going deeper
A tight-binding model keeps only two ingredients: the energy of an electron on an atom, and the amplitude to hop to a neighbour. That is enough to produce a band, and cheap enough to handle the tens of thousands of atoms in a moiré supercell, where first-principles calculation cannot follow.
Two numbers make a band
Write the electron wavefunction as a combination of orbitals sitting on atoms, keep an on-site energy ε and a hopping amplitude t to nearest neighbours, and the resulting band for a one-dimensional chain is ε − 2t cos(ka): a cosine of total width 4t. Bandwidth is set by hopping, so weakly overlapping orbitals give narrow bands and strongly overlapping ones give wide bands.
In two dimensions the same recipe on a gives graphene’s , and with different energies on the two sublattices it gives hBN’s gap. Slater and Koster showed in 1954 how to parametrise the hoppings between orbitals of any symmetry with a small set of numbers, which is what makes the approach systematic rather than ad hoc.
Why it is indispensable in 2D
handles hundreds to a few thousand atoms. A moiré supercell at the holds more than ten thousand; averaging, transport calculations and large devices need more still. Tight-binding models run on millions of atoms, which puts twisted bilayers, defect ensembles and realistic device geometries within reach.
They are also compact enough to reason with. Three bands built from the metal d orbitals reproduce the and the of a group-VI , which is enough for physics, optical selection rules and transport – with a handful of parameters instead of a full calculation for every new question.
What it cannot do
The parameters have to come from somewhere: fitted to first-principles bands, to experiment, or derived by projecting a calculation onto localised orbitals. A model fitted at one condition interpolates well and extrapolates badly – , pressure, a different or heavy can change the hoppings in ways the fit does not know about.
It also contains no self-consistency by default: charge redistribution, screening and interactions must be added by hand, for example through a term or a self-consistent Hartree potential, and each addition brings its own parameters. A tight-binding model is best understood as a compact summary of a someone else computed or measured, useful precisely because it is not a prediction from first principles.
For specialists
An expansion of the Hamiltonian in localised orbitals with hopping integrals between them, fitted to first-principles bands or constrained by symmetry in the Slater–Koster scheme. Its value in 2D is scale: moiré supercells of 104–105 atoms, disorder averaging and transport are out of reach for DFT and routine here. Three bands built from the metal d orbitals already capture the band edges and spin–orbit splitting of a group-VI TMDC monolayer.