Standing Friedel Waves, Standing Spin Waves, and Indirect Bandgap Optical Transition in Nanostructures
Jun-Qiang Lu1, X.-G. Zhang1,2, and Sokrates T. Pantelides3
for Nanophase Materials Sciences, Oak Ridge National Laboratory
Theoretical investigation of the dynamicresponse of electrons in a nanowire or a two-dimensional electron gas under a time-dependent external field reveals a standing Friedel wave near edges and boundaries. This is analogous to the static Friedel oscillations near defects at equilibrium. Furthermore, on a finite strip of graphene nanoribbon, the spin-polarized edge states can be excited optically to generate standing spin waves. The wavelengths of these standing waves are determined by the wave vectors connecting different parts of the Fermi surface, analogous to the 2kF wave vector of the static Friedel oscillations. The amplitudes of the standing waves exhibit resonances as a function of the driving frequency of the external field. These resonances provide information on the energy levels of the nanostructure. For the spin-resolved edge states of the graphene nanoribbon under a transverse static electric field, the standing spin waves on the two opposing edges have different wave lengths, corresponding to different parts of the graphene band structure.
The standing spin waves in graphene nanoribbons are optical excitations from an indirect bandgap transition. Such transitions are not possible in macroscopic bulk materials because of the well-known fact that optical waves cannot provide the momentum transfer necessary for electrons at one Bloch state to jump to another Bloch state with a different wave vector. Indirect bandgap transitions in bulk Si, for example, can only occur as a phonon assisted process. In a nanoscale system, however, these optical transitions are possible because both k and –k excitations are generated simultaneously thus producing a standing wave and conserving the momentum. At even smaller scales, e.g., in small molecules, optical transitions can occur but the small size does not allow detectable spatial waves to form. Thus the standing Friedel waves and standing spin waves are unique features of nanoscale systems.
An early part of this work was published as J.-Q. Lu, X.-G. Zhang, and S. T. Pantelides, Phys. Rev. Lett. 99, 226804 (2007).