# Generalized Electron Emission Theory for One-Dimensional Conducting Materials
## 6 Results and Discussion
The calculation of the one-dimensional current was conducted for materials with a Fermi energy of 4.57 eV and various work functions, specifically 2, 3, 4, and 5 eV. metals, because their forbidden band gap can be negligible. The field enhancement factor is defined as \(\beta=\frac{\mathrm{F}}{\mathrm{F_{0}}}\), where \(\mathrm{F_{0}}\) is the mean applied electric field and \(\mathrm{F}\) is the electric field at the tip. The field enhancement factor of CNTs can be higher than 1000, so the applied voltage is much lower than that for the conventional field emitters such as tungsten tips. For a single nanotube, the electric field enhancement factor is \(\beta=\frac{\mathrm{h}}{d}\), where \(\mathrm{d}\) and \(\mathrm{h}\) are the diameter and height of the nanotube, respectively [19]. The electric field enhancement factor can be approximated to \(\beta=\frac{\mathrm{h}}{d}(1+\frac{d}{D})\) when distance \(D\) between the nanotube tip and the anode surface is considered [19]. The electron emission experiments from CNTs basically follow the Fowler-Nordheim equation when considering logarithm data analysis [15; 27]. In this paper, we consider one-dimensional field emission rather than three-dimensional field emission. We also consider the thermionic emission effect from the heating of CNTs. This research covers the generalized electron emission effects from arbitrary electric fields and arbitrary temperatures. In Figure 5, the one-dimensional current of carbon nanotubes is presented as a function of the electric field. The work function and Fermi level function for carbon nanotubes are specified as 4.56 eV and 4.57 eV, respectively. The temperatures range from 500 K to 2500 K, progressing from bottom to top. The depicted trend reveals that the current emitted from carbon nanotubes rises with increases in both the temperature and the applied accelerating field. Figure 6 illustrates the variation in the one-dimensional current with the applied electric field for various work functions, maintaining a constant temperature of 1000 K. The depicted graph presents a sequence of work functions ranging from 2 to 5 eV, from top to bottom. The observed trend indicates that the current increases with a rise in the applied accelerating field. This research is valuable for understanding electron emissions originating from both electric potential energy and thermal energy within diverse one-dimensional materials. The calculation of the one-dimensional current elucidates thermionic emission, highlighting its temperature dependency, as well as field emission, emphasizing its sensitivity to strong electric fields. Additionally, it elucidates generalized electron emission, encompassing both effects and accommodating arbitrary variations in the temperature and electric field. Figure 5: One-dimensional current of carbon nanotubes plotted against the accelerating field. In the case of carbon nanotubes, the work function and Fermi level are specified as 4.56 eV and 4.57 eV, respectively. The represented temperatures ascend from bottom to top, representing 500, 1000, 1500, 2000, and 2500 K.