Electron states that exist only at the surface of a crystal, because the crystal stops there. Atoms at a surface have lost their neighbours on one side, so electrons can take energies there that the inside of the crystal does not allow. Usually such states are a nuisance that traps charge. In a they are the point: the inside insulates while the surface is guaranteed to conduct, with each electron’s tied to its direction of motion.
Going deeper
Where the crystal ends, electrons can take energies the bulk forbids, in states that live in the last few atomic layers. Ordinary surface states trap charge and are passivated away. In a topological insulator surface states must cross the bulk gap, forming a Dirac cone in which each electron’s spin is tied to its direction of motion.
Why a surface has states of its own
Inside a crystal an electron meets the same arrangement of atoms repeated in every direction, and that repetition decides which energies it may have. At a surface the repetition stops. Solutions of the wave equation that would grow without limit in an endless crystal become allowed when they die away from the surface into the bulk, and they give states that live only in the last few atomic layers. Igor Tamm described them in 1932 and William Shockley in 1939.
and rearranged surface atoms add more. These ordinary surface states trap charge, pin the and bend the bands at surfaces and contacts, and much of the craft of semiconductor technology lies in passivating them. Layered crystals are the exception that makes attractive: their faces have no dangling bonds, so clean surfaces are largely free of such traps.
Topological surface states
In a topological insulator the surface states cannot be removed. The bulk bands carry a invariant that differs from that of empty space, and the change at the surface forces states to cross the gap. In Bi2Se3 angle-resolved shows a single at the surface, inside a bulk gap of about 0.3 eV, and spin-resolved measurements show each electron’s spin locked at right angles to its momentum. An electron cannot reverse its direction without flipping its spin, which ordinary impurities cannot do, so scattering straight back is suppressed.
have surface states of another shape: open Fermi arcs joining the surface projections of Weyl points of opposite . In the layered Weyl semimetal PtBi2 these arcs superconduct on their own, which is one reason the material is studied.
Thin films and the 2D limit
Experiments are dominated by practical problems. make most Bi2Se3 crystals , so the bulk conducts alongside the surface and masks it; compensating dopants, gates and such as (Bi,Sb)2Te3 bring the Fermi level back into the gap. Exposure to air dopes the surface and adds bands of ordinary states on top.
Thinning sets its own limit. Below about six quintuple layers the states on the top and bottom faces of a Bi2Se3 film overlap and couple, opening a gap, and the film becomes an whose character, ordinary or topological, depends on the exact thickness. In a truly two-dimensional material the faces of the sheet are not boundaries in this sense: the boundary of a 2D topological phase is its edge, and the counterpart of the surface state is the .
For specialists
States localised at a crystal surface and decaying into the bulk, at energies forbidden in the bulk for that in-plane momentum. Ordinary Tamm and Shockley states come from the broken periodicity, dangling bonds or reconstruction, and can be removed or shifted by . Topological surface states are required by a non-trivial bulk invariant: a single spin–momentum-locked Dirac cone in Bi2Se3-family topological insulators, protected against backscattering by , and Fermi arcs joining the projections of Weyl points in Weyl semimetals. In thin films the top and bottom surface states hybridise and open a gap below a critical thickness, about six quintuple layers for Bi2Se3. In a 2D material the boundary of a topological phase is its edge, and the counterpart of the surface state is the edge state.