Definition of Graphene: Graphene is a 2-brane which anchors a triangular lattice of quarks whose vertices represent virtual locations or density of states characteristic of carbon atoms. Superimposed on this 2-brane is its chiral complement in which the triangles are rotated 180 degrees and moved, so that, each overlapping pair of triangles forms a Star of David. Sigma orbitals link the 2-branes together in a honeycomb arrangement. Pi orbitals with two valleys form like spouting mushrooms above and below the 2-brane. This permits Cooper pairs since they are time reversed electron states. The characteristic rippled shape of graphene means that the Fourier transform looks like a hexagon with light cones at the vertices rather than the cylinders it would have, if you could populate it in completely flat form. The properties of graphene are highly variable depending on how the 2-brane is populated internally and on the boundaries.
The total magnetic moment of graphene is zero when there are equal numbers of carbon atoms in its 2-brane and chiral complement. In the general case the total magnetic moment is half the difference between the numbers of atom populating the 2-branes. The magnetization in a zigzag edge has ferromagnetic ordering on each edge, but anti-ferromagnetic coupling between the edges. A single layer of graphene under low magnetic field oscillations has a Berry phase of pi while two layer systems display a Berry phase of 2pi. Similarly, a single layer has almost negligible magnetoresistance, while more than one layer has more conventional properties. Two layer graphene is most notable with respect to its noise cancelling properties.
Internal and external migration of carbon atoms forming pentagons and heptagons can warp the shape of the 2-brane without affecting the threefold coordination of the carbon atoms.
When a single carbon atom is missing from an internal hexagon, the resulting pentagon has “weak” and “gravitational” effects:
1. It exchanges the amplitudes of the two lattices.
2. It exchanges the two Fermi points.
3. It induces a phase coupling proportional to the deficit angle.
4. It induces local (4nm) “gravitational” curvature in spacetime and thus in the sheet. Spin connections provide a generalization of gauge fields used to model spacetime.
The resulting heptagon is paired with a pentagon. The pair acts as a diplole. The pentagon attracts charge and the heptagon repels charge. The heptagon induces negative “gravitational” curvature.
The total magnetic moment of graphene is zero when there are equal numbers of carbon atoms in its 2-brane and chiral complement. In the general case the total magnetic moment is half the difference between the numbers of atom populating the 2-branes. The magnetization in a zigzag edge has ferromagnetic ordering on each edge, but anti-ferromagnetic coupling between the edges. A single layer of graphene under low magnetic field oscillations has a Berry phase of pi while two layer systems display a Berry phase of 2pi. Similarly, a single layer has almost negligible magnetoresistance, while more than one layer has more conventional properties. Two layer graphene is most notable with respect to its noise cancelling properties.
Internal and external migration of carbon atoms forming pentagons and heptagons can warp the shape of the 2-brane without affecting the threefold coordination of the carbon atoms.
When a single carbon atom is missing from an internal hexagon, the resulting pentagon has “weak” and “gravitational” effects:
1. It exchanges the amplitudes of the two lattices.
2. It exchanges the two Fermi points.
3. It induces a phase coupling proportional to the deficit angle.
4. It induces local (4nm) “gravitational” curvature in spacetime and thus in the sheet. Spin connections provide a generalization of gauge fields used to model spacetime.
The resulting heptagon is paired with a pentagon. The pair acts as a diplole. The pentagon attracts charge and the heptagon repels charge. The heptagon induces negative “gravitational” curvature.
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