# Generalized Electron Emission Theory for One-Dimensional Conducting Materials
## 6 Results and Discussion  
Figure 1 depicts a schematic representation of one-dimensional electron emission. A robust local electric field is directed along the \(x\)-axis within the metal. At zero temperature, electrons occupy energy levels up to the Fermi energy. As the temperature rises, these electrons become excited, reaching energy levels exceeding the Fermi energy and, subsequently, surpassing the work function. Thermally excited electrons, whose energy levels exceed that of the vacuum, participate in thermionic emissions. On the other hand, under a strong electric field, electrons with energy levels below the vacuum energy engage in quantum tunneling, constituting the generalized field emission. The one-dimensional materials can be nanotubes or nanowires. For one-dimensional electron emission, the Fermi energy and work functions are important parameters. The interactions between electrons and phonons are considered small relative to the electron emission effect, so they are neglected in this research. Figure 2 depicts the spectral radiance for various temperatures in one-dimensional materials, presented in terms of dimensionless photon energy. The spectral radiance, as defined in Equation (9), is expressed as dimensionless photon energy (\(\frac{h\
u}{k_{B}T}\)) for various temperatures in one-dimensional materials. The temperature sequence in Figure 2, progressing from bottom to top, corresponds to 2.7 K, 300 K, 1000 K, 3000 K, and 10,000 K. It is noteworthy that the unit of the one-dimensional energy density is joules per meter (J/m), while the unit for the three-dimensional energy density is joules per cubic meter (J/m). In Figure 3, the variation in the one-dimensional current with temperature is presented for different materials characterized by distinct work functions. The calculation of the one-dimensional current density was performed for materials with a Fermi energy of 4.57 eV and varying work functions from 2 to 5 eV. The observed trend indicates that the one-dimensional current density rises with increasing temperature and decreasing work function. This calculation for one-dimensional materials is consistent with well-known thermionic emission results. Figure 2: Spectral radiance presented as dimensionless photon energy for various temperatures in one-dimensional materials. The spectral radiance, as defined in Equation (9), is represented with dimensionless photon energy (\(\frac{h\
u}{k_{B}T}\)) across different temperatures in one-dimensional materials. The sequence of temperatures, from bottom to top, corresponds to 2.7, 300, 1000, 3000, and 10,000 K.  
Figure 4 illustrates the one-dimensional current plotted against the electric field for various materials characterized by various work functions. The calculation of the one-dimensional current was conducted for materials with a fixed Fermi energy of 4.57 eV and varying work functions ranging from 2 to 5 eV. It was observed that the one-dimensional current of the conducting material increases with an increase in the electric field and a decrease in the work function. This calculation for one-dimensional materials is consistent with well-known field emission results. One-third of single-walled nanotubes (SWNTs) are good conductors because their armchair structure shows conducting properties, and the rest show semiconducting properties [19]. Multi-walled nanotubes (MWNTs), which consist of many SWNTs, are considered  
Figure 4: One-dimensional current plotted against the accelerating field for materials with varying work functions. The temperature is zero, and only field emission is considered. The calculation involved determining the one-dimensional current for materials with a common Fermi energy of 4.57 eV and different work functions, specifically 2, 3, 4, and 5 eV. Figure 3: One-dimensional current with temperature for various materials characterized by distinct work functions. The electric field is zero, and only thermionic emission is considered.