A flat layer of atoms arranged in squares, like the corners of the squares on a chessboard, each atom with four neighbours. Such sheets sit inside many layered crystals – iron in the FeSe , tellurium in the rare-earth tritellurides, silicon in ZrSiS – and because their electrons spread across the whole sheet, they largely decide how the crystal conducts. A perfect square net is often restless: its atoms can pair up or its electrons form a , and where it stays square, its bands can cross to give fast, nearly electrons.
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The tellurium square nets have a nearly nested Fermi surface, so the electrons and lattice lock into a periodic modulation that in the lightest members survives above room temperature.
A square net of atoms conducts across the whole sheet, but is prone to distort: pairing up the atoms lowers the energy by opening a gap over part of the Fermi surface.
Where square nets are found
Square nets are among the most common motifs in layered crystals. In FeSe and the iron a square net of iron atoms, each inside a of chalcogen or pnictogen atoms, carries the superconductivity. In the rare-earth tritellurides RTe3, each layer holds two flat square nets of tellurium on either side of a puckered RTe slab. In ZrSiS, ZrSiSe and their relatives a silicon or germanium net lies between layers of zirconium and sulfur, and in compounds such as SrMnBi2 and EuMnBi2 the net is bismuth.
Many of these crystals cleave easily along the layers, and several, including FeSe, the tritellurides and ZrSiS, have been thinned to a or grown as thin films.
Why the squares are restless
In a square net of main-group atoms each atom has four neighbours, more than its electrons can serve with ordinary two-electron bonds, so the bonding is shared out over the net in weaker bonds – chemists call it hypervalent. The partly filled p bands that result have flat, parallel stretches of , and such nesting invites a distortion that opens a gap and lowers the energy, much as a Peierls distortion does in a chain of atoms.
Depending on the electron count, the atoms pair up into chains, ladders or zigzag patterns, or a gentle, charge density wave forms. The tritellurides show the second: for most of the rare earths a charge density wave sets in above room temperature, and squeezing the lattice by swapping in smaller rare-earth ions lowers the transition, until the heaviest show a second wave at low temperature.
Square nets and Dirac electrons
When a square net stays square, it can host electrons that behave as if massless. The layers above and below double the repeating unit of the net, which folds its p bands back on themselves so that they cross near the . A glide symmetry of the crystal – a reflection combined with a half-step shift – protects some of these crossings from opening a gap. ZrSiS, studied by in 2016, has a whole network of such crossings, , with bands that stay straight over an unusually wide energy range.
This gives chemists a recipe: look for square nets that the electron count and keep undistorted, and expect Dirac or nodal-line bands. Bismuth and antimony nets add strong , and in AMnBi2 and AMnSb2 the neighbouring manganese layers add magnetism, so that the Dirac electrons and the can act on each other.
For specialists
A planar net of atoms with fourfold coordination, a recurring building block of layered compounds. In square nets of main-group atoms such as Si, Sb, Bi or Te the bonding is electron-rich and delocalised, the p bands are partly filled and well nested, and the net tends to distort – into chains, ladders or zigzag patterns, or the incommensurate charge density waves of the RTe3 series – unless the electron count favours it staying square. When it does, the doubling of the by the layers above and below folds the p bands so that they cross near the Fermi level, and a nonsymmorphic glide symmetry protects the crossings: ZrSiS and its relatives are nodal-line semimetals, and AMnBi2 and AMnSb2 combine such Dirac bands with magnetism. Transition-metal square nets, as in FeSe and the CuO2 planes of the cuprates, carry the d-electron physics of the iron-based and cuprate superconductors.