Magnetization of the metallic surface states in topological insulators.

We calculate the magnetization of the helical metallic surface states of a topological insulator. We account for the presence of a small sub-dominant Schrödinger piece in the Hamiltonian in addition to the dominant Dirac contribution. This breaks particle-hole symmetry. The cross-section of the upper Dirac cone narrows while that of the lower cone broadens. The sawtooth pattern seen in the magnetization of the pure Dirac limit as a function of chemical potential (µ) is shifted; but, the quantization of the Hall plateaus remains half integral. This is verified by taking the derivative of the magnetization with respect to µ. We compare our results with those when the non-relativistic piece dominates over the relativistic contribution and the quantization is integral. Analytic results for the magnetic oscillations are obtained where we include a first order correction in the ratio of non-relativistic to relativistic magnetic energy scales. Our fully quantum mechanical derivations confirm the expectation of semiclassical theory except for a small correction to the expected phase. There is a change in the overall amplitude of the magnetic oscillations. The Dingle and temperature factors are modified.

Valley-spin polarization in the magneto-optical response of silicene and other similar 2D crystals.

We calculate the magneto-optical conductivity and electronic density of states for silicene, the silicon equivalent of graphene, and similar crystals such as germanene. In the presence of a perpendicular magnetic field and electric field gating, we note that four spin- and valley-polarized levels can be seen in the density of states, and transitions between these levels lead to similarly polarized absorption lines in the longitudinal, transverse Hall, and circularly polarized dynamic conductivity. While previous spin and valley polarization predicted for the conductivity is only present in the response to circularly polarized light, we show that distinct spin and valley polarization can also be seen in the longitudinal magneto-optical conductivity at experimentally attainable energies. The frequency of the absorption lines may be tuned by the electric and magnetic field to onset in a range varying from THz to the infrared. This potential to isolate charge carriers of definite spin and valley label may make silicene a promising candidate for spin- and valleytronic devices.