Novel spin-orbit coupling in the absence of local inversion symmetry

Normally we associate spin-orbit coupling with degenerate atomic orbitals (or energy bands) associated with d- or f-orbitals. However, in solid state physics a quite distinct type of spin-orbit coupling can occur and has attracted a lot of interest over the past decade.

In a seminal 2005 paper [which took 12 months for PRL to publish!] Kane and Mele proposed that in graphene a spin quantum Hall effect could occur due to spin-orbit coupling. Moreover, this paper proposed that this state was a topological insulator, starting a whole industry. I want to just focus on the spin-orbit coupling term in the Hamiltonian that is the first step in their argument.

This term arises because there are two carbon atoms per primitive unit cell in the crystal lattice. [A and B sub lattice]. It does not have local inversion symmetry.

How large is Delta_so ?
Kane and Mele estimated, based on a crude argument, that is was about 1.2 Kelvin. But, then they gave a renormalisation group argument, claiming that electron-electron interactions would increase the value to something like 7.5 K.
However, much more sophisticated analysis, such as this one, showed that Delta_so arose from subtle pi-sigma orbital mixing and was orders of magnitude smaller! Hence, the chance of seeing a quantum spin Hall effect in graphene are extremely unlikely.

Aside. This illustrates you can be wrong about something but still stimulate a whole new field. But, in fairness, everything is correct about the paper, except the parameter estimate for graphene. This is quite different to people who publish papers that are just plain wrong, but still stimulate positive outcomes.

What about other systems?
A nice example is monolayer MoS2, as discussed here.

A full three-dimensional crystal of MoS2 has inversion symmetry. However, a monolayer does not.
If you take a Mo atom as an inversion centre, a S atom is mapped onto an empty location.
Delta_so is estimated to be about 500 K.
It is orders of magnitude larger than graphene because the bare-spin orbit coupling is much larger due to the heavy Mo atoms.

A similar spin-orbit coupling has been proposed to occur in a quasi-one-dimensional metal, Li0.9Mo6O17.