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Optical spectral weight: comparison of weak and strong spin-orbit coupling

The Fermi velocity ($v_{F}$) associated with the spin-orbit coupling is two orders of magnitude smaller for spintronic semiconductors than it is for topological insulators. Both families can be treated with the same Hamiltonian which contains a relativistic (Dirac) linear in momentum term proportional to $v_{F}$ and a non-relativistic quadratic contribution with Schrödinger mass (m). We find that the AC dynamic longitudinal and transverse (Hall) magneto-conductivities are strongly dependent on the size of $v_{F}$. When the Dirac fermi velocity is small, the absorption background provided by the interband optical transitions is finite only over a very limited range of photon energies as compared with topological insulators. Its onset depends on the value of the chemical potential ($μ$) and on the magnetic field (B), as does its upper cut off. Within this limited range its magnitude is however constant and has the same magnitude of $e^{2}π/(8h)$ as is found in topological insulators and also in graphene noting a difference in degeneracy factor. The total optical spectral weight under the universal interband background is $e^{2}π/(8h)4mv_{F}^{2}$. In contrast to the known result for graphene no strict conservation law applies to the spectral weight transfers between inter and intra band transition brought about by variations in the magnitude of the chemical potential when a non-relativistic contribution is present in the Hamiltonian whatever size it may have.