Exact Landau spectrum and wave functions of biased AA-stacked multilayer graphene

A novel model is presented to obtain exact energy spectrum, Landau levels (LLs) and wave functions in AA-stacking N-layer Graphene under the influence of perpendicular electric and magnetic fields. A particular unitary transformation to deal with the broken mirror symmetry caused by a perpendicular electric field can transform an AA-stacking N-layer graphene into N independent graphene-like layers. The exact energy spectrum of a graphene-like layer is equal to E=EMLG+ε, where EMLG is the energy spectrum of a monolayer graphene and ε is controlled by the magnitude of electric field and layer number. The associated wave functions are clearly specified. Such a method is applied to analytically study the combined effect of the vertical electric field and magnetic fields on LLs and wave functions in AA-stacking N-layer graphenes. An analytical formula of the gate-tuned LLs clearly exhibits the dependence of LLs on the number of layers, strength of gated field and magnetic fields.