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Graphene-based allotropes such as carbon nanorings hold the promise of
completely new nanodevice and metamaterials applications due to the
effects of magnetic flux and curvature on quantum transport on a
nanoscale toroidal surface and the coherence of resulting
electromagnetic moments. Unique electronic and optical characteristics
will emerge due to the compactification of the honeycomb lattice
structure of a flat graphene sheet to a two-diemsional mamifold with
toroidal geometry. Additional modular symmetries are predicted to
significantly impact energy band structure and transport properties of
physically distinct nanotori with different chiralities and dimensions
and thus drastically reduce the number of spectrally distinct ring
geometries. In addition to persistent current and Aharonov-Bohm effects
under magnetic flux, new electromagnetic field distributions such as a
new toroidal moment will be generated by the ring currents. In a
metamaterial of a regular two- or three-dimensional lattice of these
aligned nanoconstituents a significant enhancement of these quantum
signatures may be expected coherence of the individual electromagnetic
responses. In an effective model, the Hamiltonian for a single charge
constrained to motion near a toroidal helix with loops of arbitrary
eccentricity is developed and the resulting three-dimensional Schrödinger
equation reduced to an effective one-dimensional formula inclusive of
curvature effects in form of two resulting effective curvature
potentials. The magnitude of the toroidal moment generated by the
current depends strongly on the magnetic field component of the field
normal to the toroidal plane. A strong dependence on coil eccentricity
is also observed. In a theoretical sense, the curvature potential terms
are necessary to preserve the hermiticity of the minimal prescription
Hamiltonian. This effective model may also elucidate how a surface
current may be driven by a properly polarized incoming electromagnetic
wave front to generate a specific multipole response. Alternatively,
electron transport on the carbon nanotorus is calculated in a
tightbinding model for armchair and zigzag carbon nanotori between
metallic leads using a recursive non-equilibrium Green's function
method. Density-of-states, transmission function and source drain
current are calculated for realistic system sizes of 10,000 carbon atoms
and more. An object-oriented C++ code was developed using parallel
sparse matrix software libraries such as PETSc (Portable, Extensible
Toolkit for Scientific Computation) with additional MPI parallelism to
evaluate the transport Green.s function at different energies. This fast
and numerically precise tool on a multi-core architecture can
incorporate additional effects such as electron-phonon coupling effects
due to low-energy phonon modes, exciton transport, or electron-plasmon
coupling terms in second- or third-nearest-neighbor type calculations.
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