Mirror symmetry

Everyday term

In plain words

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’ behave, which band crossings are protected, and even how a calculation of it has to be set up.

Going deeper

For a single layer the mirror that matters most is the one in the plane of the sheet: does it look the same from above and below? If it does, spins split by spin–orbit coupling point straight out of the plane and no dipole can exist across the layer. Break it and both statements fail.

The horizontal mirror

A has a mirror plane through its metal layer: the planes above and below are identical, so reflecting the layer leaves it unchanged. That single symmetry has several consequences. It forbids any electric dipole across the layer, so the two faces are electrically equivalent. And when combined with the lack of an , it constrains to point spins perpendicular to the layer, out of plane, with opposite directions in the two – the locking behind and the valley-dependent spin splitting.

Remove the mirror – with a layer that has sulfur on one face and selenium on the other, with a , or with a gate field – and spins acquire in-plane components: physics appears, and so does a dipole.

A practical consequence for calculations

The mirror also matters to anyone running a periodic calculation. A slab without the horizontal mirror carries a dipole across it, and in a periodic supercell that dipole interacts with its own images through the vacuum, shifting the potential and the band energies and converging slowly with vacuum size.

The standard remedy is a dipole correction: an artificial sheet of charge placed in the vacuum that cancels the field. For symmetric slabs it is unnecessary, which is why the issue is easy to forget – and easy to get wrong when the model is changed to a Janus layer, a layer on a substrate or a gated slab. Published numbers for asymmetric slabs should say whether the correction was applied.

Vertical mirrors and protected crossings

Mirrors perpendicular to the layer matter too, because they can protect band degeneracies. States on a mirror plane can be labelled by their mirror eigenvalue, and two bands with different labels may cross without repelling. Whole lines of such crossings give – PbTaSe₂ is a layered example – and a mirror-graded invariant, the mirror , classifies topological crystalline such as the SnTe family.

Because these protections rest on a crystalline symmetry rather than on time reversal, they are fragile in a specific way: , a substrate or a surface reconstruction that breaks the mirror gaps the crossings out. That sensitivity cuts both ways, since it also makes such states tunable by strain.

For specialists

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 .

Where this comes from

Topological crystalline insulators in the SnTe material class Hsieh et al. · Nature Communications 3, 982 (2012) cited by 1,453
Topological nodal-line fermions in spin-orbit metal PbTaSe₂ Bian et al. · Nature Communications 7, 10556 (2016) cited by 860

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

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

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

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

Rashba effect

When a crystal is not the same seen from above and from below – because it sits on a substrate, has a field across it, or is simply built that way – a moving electron feels a magnetic field that depends on which way it is going. Its spin then winds around its direction of motion, which is what lets an electric field steer spins.

A momentum-dependent spin splitting from spin–orbit coupling in a structure without inversion symmetry, H = α(σ × k)·ẑ, locking spin perpendicular to in-plane momentum and winding it around the Fermi contour. The coefficient α follows the potential gradient and the atomic spin–orbit strength, so it is tunable by gating, by the substrate and by the choice of heavy elements; BiTeI splits its bands far enough for the two spin-split Fermi surfaces to be resolved separately.

Full explanation

Weyl and Dirac semimetal

Crystals in which the highest filled band of electron energies and the lowest empty one touch only at isolated points, instead of overlapping broadly or leaving a gap between them. Electrons near such a point behave as if they had no mass – the crystal is a three-dimensional cousin of graphene – and its surface carries loose-ended arcs of electron states that no ordinary metal can have.

Semimetals whose bands touch at isolated points with linear dispersion in every direction. A Dirac point is fourfold degenerate and protected by time-reversal and inversion symmetry together; breaking either splits it into pairs of twofold Weyl nodes of opposite chirality, which act as sources and sinks of Berry curvature and are joined across the surface by Fermi arcs. Where a mirror or another crystalline symmetry protects a whole line of band touchings instead of isolated points, the same physics gives a nodal-line semimetal with drumhead surface states. The chiral anomaly, large anomalous Hall response and strongly non-linear optical effects follow from that structure.

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

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

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