We consider a theory of a two-component Dirac fermion localized on a (2+1) dimensional brane coupled to a (3+1) dimensional bulk. Using the fermionic particle-vortex duality, we show that the theory has a strong-weak duality that maps the coupling e toẽ = (8π)/e. We explore the theory at e 2 = 8π where it is self-dual. The electrical conductivity of the theory is a constant independent of frequency. When the system is at finite density and magnetic field at filling factor ν = 1 2 , the longitudinal and Hall conductivity satisfies a semicircle law, and the ratio of the longitudinal and Hall thermal electric coefficients is completely determined by the Hall angle. The thermal Hall conductivity is directly related to the thermal electric coefficients.

We demonstrate the syntheses of single crystalline Cu nanobelt and Ag belt-like materials via CTAC-assisted (CTAC, cetyltrimethylammonium chloride) galvanic reductions. The single crystalline face-centered cubic phase Cu nanobelt was prepared by reacting CuCl2(aq) with Al(s) in an aqueous solution of CTAC and HNO3. The Cu nanobelt exhibited a high-quality ribbon-like nanostructure with a thickness less than 15 nm, a width of 30-150 nm, and a length up to 10 microm. The belt-like Ag, with a thickness less than 10 nm, a width of 30-100 nm, a length up to 5 microm, and a novel single crystalline 4H structure, was prepared by reacting AgNO3(aq) and Cu(s) in an aqueous solution of CTAC.

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Urchin-like silver nanowires are prepared by reacting AgNO(3)(aq) with copper metal in the presence of cetyltrimethylammonium chloride and HNO(3)(aq) on a screen-printed carbon electrode at room temperature. The diameters of the nanowires are about 100 nm, and their lengths are up to 10 μm. Using Raman spectroscopy, the detection limit of Rhodamine 6G (R6G) on the urchin-like silver nanowire substrate can be as low as 10(-16) M, while the analytical enhancement factor is about 10(13). Raman mapping images confirm that a single R6G molecule on the substrate can be detected.