The dispersion relation of periodic structures that include metamaterials or materials with large anomalous dispersion can give bands with infinite group velocity points. These bands do not span the entire first Brillouin zone but are instead localized in k-space. We show that these points arise when both positive and negative elements are present, with the group index rather than the refractive index being the controlling quantity. A rigorous condition and two approximations are derived, each showing that an appropriate weighted average of group index being zero leads to infinite group velocity points.
Electron field emission from a single nanoemitter is a barrier tunneling, quantum mechanical process that can, therefore, be described by the well-known Fowler-Nordheim (FN) equation. At high emission current densities, however, the space charge caused by the cathode may affect the current density-voltage (J-V) characteristics predicted by the FN theory. In this study, we theoretically investigated the effect of space charge on FE nanodevices, including diode and triode structures. The J-V characteristics of FE nanodevices were obtained by analytically (diode structures) or numerically (triode structures) solving the coupled FN equation and Poisson's equation. We discuss the behavior of FE nanodiodes and nanotriodes displaying different geometries, dimensions and work functions of their emitter materials. In the high current density region, space charge plays an important role in FE nanodevices; the threshold current density of space-charge limitation is related to the electric field distributions. Besides, our theoretical results are in good agreement with the experimental results reported previously.
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