High magnetic field theory for the local density of states in graphene with smooth arbitrary potential landscapes
Résumé
We study theoretically the energy and spatially resolved local density of states (LDoS) in graphene at high perpendicular magnetic field. For this purpose, we extend from the Schrödinger to the Dirac case a semicoherent-state Green's function formalism, devised to obtain in a quantitative way the lifting of the Landau level degeneracy in the presence of smooth confinement or disordered potentials. Our general technique, which rigorously describes quantum mechanical motion in a magnetic field beyond the semi-classical guiding center picture of vanishing magnetic length (both for the ordinary two-dimensional gas and graphene), is connected to the deformation (Weyl) quantization theory in phase space developed in mathematical physics. For generic quadratic potentials of either scalar (i.e., electrostatic) or mass (i.e., associated with coupling to the substrate) types, we exactly solve the limit of large cyclotron frequency (yet at finite magnetic length) where Landau level mixing becomes negligible. Hence, we obtain a closed-form expression for the graphene Green's function in this regime, providing analytically the discrete energy spectra for both cases of scalar and mass parabolic confinement. Furthermore, the coherent-state representation is shown to display a hierarchy of local energy scales ordered by powers of the magnetic length and successive spatial derivatives of the local potential, which allows one to devise controlled approximation schemes at finite temperature for arbitrary and possibly disordered potential landscapes. As an application, we derive general analytical non-perturbative expressions for the LDoS, which may serve as a good starting point for interpreting experimental studies. For instance, we are able to account for many puzzling features of the LDoS recently observed by high magnetic field scanning tunneling spectroscopy experiments on graphene, such as the \sqrt{m}-increase of the m^th Landau level width in the LDoS peaks, together with a flattening of the spatial variations of the Landau level energies at increasing m.