Boundary matters: topological insulators

It has become a habit for those who profess in solid state physics to consider a crystal as a periodic array of atoms. In reality, this is, however only an approximation. Any solid is limited by its surfaces, which means the periodicity terminates at these bounds. Despite this, people still in most cases take them as infinite and unbounded, so that exact periodicity can be used to obtain exact solutions of some models. Usually, such solutions indeed provide very good descriptions of the sample, provided the bulk is dominant over the boundaries. One such artificial boundary condition goes under 'Von Karmen periodic condition', which yields Bloch waves.

Nevertheless, surface (not film, [1]) states can display many exotic properties that are not supported by bulky solutions. These properties may be related to, let's say, impurities, dangling bonds, surface tensions, structural reconstructions and et al. Due to these properties, modern computers could be made. As we know, transistors and diodes just make use of the properties of interfaces between two semiconductors.

Surface states have not ceased to surprise people. Some ten years ago, people found a novel type of conducting channels in two dimensional electron gas systems. Such systems under strong magnetic field exhibit the famous quantum Hall effects. It was later suggested that, such Hall states possess edge states that can conduct electricity along the edges of the 2D sample. These states are squeezed out of and split from the insulating bulk states, by magnetic field.

In 2005, Kane and his collaborators suggested that [2], such edge states could exist even without magnetic field. The considered graphene, namely a single graphite layer. In such materials, they guessed, there might be strong spin-orbit coupling. Such couplings can actually play a similar role as a magnetic field, and thus may create edge states. In this case, new complexity arises due to the spin degrees of freedom (see the figure).

All the as-described edge states are 1D objects. Most recently, a kind of 2D edge states was discovered existing in Bi2Te3 compounds. Such compounds have complex crystal structures and strong spin orbit coupling. These states are able to conduct spin and charges along the surface. So, one has this very gorgeous phenomenon: insulating bulk+metallic surface. They are termed 'topological insulators'. Why topology ? Topological properties are the properties that are invariant under continuous transformations of parameter space. In Bi2Te3, that is the number of edge states, which is conserved, however the shape of the sample is changed.

Although something has been learned of these new properties, a realistic and analytically tractable solution still awaits to show up. More experiments need be done to confirm and explore our understanding.

[1]A surface is linked with a bulk, but a film has its own bulk and surface (edges). Interface can be deemed as a special surface.
[2]PRL 95, 226801 (2005)